Method for determining stack configuration of substrate

ABSTRACT

A method for determining a stack configuration for a substrate subjected to a patterning process. The method includes obtaining (i) measurement data of a stack configuration with location information on a printed substrate, (ii) a substrate model configured to predict a stack characteristic based on a location of the substrate, and (iii) a stack map including a plurality of stack configurations based on the substrate model. The method iteratively determines values of model parameters of the substrate model based on a fitting between the measurement data and the plurality of stack configurations of the stack map, and predicts an optimum stack configuration at a particular location based on the substrate model using the values of the model parameters.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority of U.S. application 62/675,918 whichwas filed on May 24, 2018, and which is incorporated herein in itsentirety by reference.

TECHNICAL FIELD

The description herein relates generally to apparatus and methods of apatterning process and determining stack configuration and/or metrologytarget design.

BACKGROUND

A lithographic projection apparatus can be used, for example, in themanufacture of integrated circuits (ICs). In such a case, a patterningdevice (e.g., a mask) may contain or provide a pattern corresponding toan individual layer of the IC (“design layout”), and this pattern can betransferred onto a target portion (e.g. comprising one or more dies) ona substrate (e.g., silicon wafer) that has been coated with a layer ofradiation-sensitive material (“resist”), by methods such as irradiatingthe target portion through the pattern on the patterning device. Ingeneral, a single substrate contains a plurality of adjacent targetportions to which the pattern is transferred successively by thelithographic projection apparatus, one target portion at a time. In onetype of lithographic projection apparatuses, the pattern on the entirepatterning device is transferred onto one target portion in one go; suchan apparatus is commonly referred to as a stepper. In an alternativeapparatus, commonly referred to as a step-and-scan apparatus, aprojection beam scans over the patterning device in a given referencedirection (the “scanning” direction) while synchronously moving thesubstrate parallel or anti-parallel to this reference direction.Different portions of the pattern on the patterning device aretransferred to one target portion progressively. Since, in general, thelithographic projection apparatus will have a reduction ratio M (e.g.,4), the speed F at which the substrate is moved will be 1/M times thatat which the projection beam scans the patterning device. Moreinformation with regard to lithographic devices as described herein canbe gleaned, for example, from U.S. Pat. No. 6,046,792, incorporatedherein by reference.

Prior to transferring the pattern from the patterning device to thesubstrate, the substrate may undergo various procedures, such aspriming, resist coating and a soft bake. After exposure, the substratemay be subjected to other procedures (“post-exposure procedures”), suchas a post-exposure bake (PEB), development, a hard bake andmeasurement/inspection of the transferred pattern. This array ofprocedures is used as a basis to make an individual layer of a device,e.g., an IC. The substrate may then undergo various processes such asetching, ion-implantation (doping), metallization, oxidation,chemo-mechanical polishing, etc., all intended to finish off theindividual layer of the device. If several layers are required in thedevice, then the whole procedure, or a variant thereof, is repeated foreach layer. Eventually, a device will be present in each target portionon the substrate. These devices are then separated from one another by atechnique such as dicing or sawing, whence the individual devices can bemounted on a carrier, connected to pins, etc.

Thus, manufacturing devices, such as semiconductor devices, typicallyinvolves processing a substrate (e.g., a semiconductor wafer) using anumber of fabrication processes to form various features and multiplelayers of the devices. Such layers and features are typicallymanufactured and processed using, e.g., deposition, lithography, etch,chemical-mechanical polishing, and ion implantation. Multiple devicesmay be fabricated on a plurality of dies on a substrate and thenseparated into individual devices. This device manufacturing process maybe considered a patterning process. A patterning process involves apatterning step, such as optical and/or nanoimprint lithography using apatterning device in a lithographic apparatus, to transfer a pattern onthe patterning device to a substrate and typically, but optionally,involves one or more related pattern processing steps, such as resistdevelopment by a development apparatus, baking of the substrate using abake tool, etching using the pattern using an etch apparatus, etc.

As noted, lithography is a central step in the manufacturing of devicesuch as ICs, where patterns formed on substrates define functionalelements of the devices, such as microprocessors, memory chips, etc.Similar lithographic techniques are also used in the formation of flatpanel displays, micro-electro mechanical systems (MEMS) and otherdevices.

As semiconductor manufacturing processes continue to advance, thedimensions of functional elements have continually been reduced whilethe amount of functional elements, such as transistors, per device hasbeen steadily increasing over decades, following a trend commonlyreferred to as “Moore's law”. At the current state of technology, layersof devices are manufactured using lithographic projection apparatusesthat project a design layout onto a substrate using illumination from adeep-ultraviolet illumination source, creating individual functionalelements having dimensions well below 100 nm, i.e. less than half thewavelength of the radiation from the illumination source (e.g., a 193 nmillumination source).

This process in which features with dimensions smaller than theclassical resolution limit of a lithographic projection apparatus areprinted, is commonly known as low-k₁ lithography, according to theresolution formula CD=k₁×λ/NA, where λ is the wavelength of radiationemployed (currently in most cases 248 nm or 193 nm), NA is the numericalaperture of projection optics in the lithographic projection apparatus,CD is the “critical dimension”—generally the smallest feature sizeprinted—and k₁ is an empirical resolution factor. In general, thesmaller k₁ the more difficult it becomes to reproduce a pattern on thesubstrate that resembles the shape and dimensions planned by a designerin order to achieve particular electrical functionality and performance.To overcome these difficulties, sophisticated fine-tuning steps areapplied to the lithographic projection apparatus, the design layout, orthe patterning device. These include, for example, but not limited to,optimization of NA and optical coherence settings, customizedillumination schemes, use of phase shifting patterning devices, opticalproximity correction (OPC, sometimes also referred to as “optical andprocess correction”) in the design layout, or other methods generallydefined as “resolution enhancement techniques” (RET). The term“projection optics” as used herein should be broadly interpreted asencompassing various types of optical systems, including refractiveoptics, reflective optics, apertures and catadioptric optics, forexample. The term “projection optics” may also include componentsoperating according to any of these design types for directing, shapingor controlling the projection beam of radiation, collectively orsingularly. The term “projection optics” may include any opticalcomponent in the lithographic projection apparatus, no matter where theoptical component is located on an optical path of the lithographicprojection apparatus. Projection optics may include optical componentsfor shaping, adjusting and/or projecting radiation from the sourcebefore the radiation passes the patterning device, and/or opticalcomponents for shaping, adjusting and/or projecting the radiation afterthe radiation passes the patterning device. The projection opticsgenerally exclude the source and the patterning device.

SUMMARY

According to an embodiment of the present disclosure, there is provideda method for determining a stack configuration for a substrate subjecteda patterning process. The method includes obtaining (i) measurement dataof a stack configuration with location information on a printedsubstrate, (ii) a substrate model configured to predict a stackcharacteristic based on a location of the substrate, and (iii) a stackmap including a plurality of stack configurations based on the substratemodel; determining, by a hardware computer system, values of modelparameters of the substrate model based on a fitting between themeasurement data and the plurality of stack configurations of the stackmap; and predicting, by the hardware computer system, an optimum stackconfiguration at a particular location based on the substrate modelusing the values of the model parameters.

According to an embodiment, the substrate model includes one or moremodels corresponding to the stack characteristic of one or more layersof the substrate.

According to an embodiment, the substrate model is expressed inCartesian coordinates having a first set of model parameters, and/or inpolar coordinates having a second set of model parameters.

According to an embodiment, the second set of model parameters isassociated with Zernike polynomials.

According to an embodiment, the stack configuration comprises aplurality of layers of the substrate, wherein each layer is associatedwith the stack characteristics.

According to an embodiment, the stack characteristic is a thickness of alayer of the substrate, a critical dimension of a feature of thesubstrate, and/or a distance between adjacent features of the substrate.

According to an embodiment, the stack characteristic is a difference ina thickness of a layer and a selected thickness of the layer.

According to an embodiment, the determining the values of the modelparameters of the substrate model is an iterative process, an iterationincludes generating the stack map having the plurality of stackconfigurations based on simulation of the substrate model and apatterning process; predicting intermediate values of model parametersbased on an optimization algorithm; and fitting the measurement data andthe plurality of stack configurations of the stack map such that a costfunction is reduced.

According to an embodiment, the patterning process comprises a designfor control process configured to automatically predict the stackconfiguration using the substrate model as perturbations.

According to an embodiment, the measurement data comprises a metrologyrecipe used for measurement of one or more stack characteristics of thestack configuration at the particular location on the substrate.

According to an embodiment, the method further includes convertingmeasurement data from a Cartesian coordinates to polar coordinates usingZernike based conversion model.

Furthermore, according to an embodiment of the present disclosure, thereis provided a method for determining optimum values of model parametersof a model configured to predict a characteristic of a patterningprocess. The method includes steps for obtaining (i) initial valuesincluding a starting point and a search region of the model parameters,(ii) measurement data corresponding to the characteristic of thepatterning process, (iii) a predicted characteristic using the initialvalues of the model parameter and the measurement data, and (iv) anobjective function, wherein the objective function comprises a firstterm related to a fit level, and a second term representing a penalty;and determining, by a hardware computer system, the values of the modelparameter based on the starting point, the search region, the fit levelbetween the model and the measurement data such that the objectivefunction is reduced.

According to an embodiment, the characteristic of the patterning processis a stack characteristic.

According to an embodiment, the stack characteristic is a substratethickness, a thickness deviation, an overlay, and/or an alignment.

According to an embodiment, the model is a substrate model representingthe stack characteristic.

According to an embodiment, the substrate model has a parabolic form.

According to an embodiment, the search region is defined by a radiuswith the starting point as a center, wherein the radius is a distancefrom a center.

According to an embodiment, the fit level is a difference between apredicted characteristic and the measurement data.

According to an embodiment, the determining the values of the modelparameter is an iterative process, wherein an iteration includesdetermining a number of sample points to be selected from the searchregion based on a number of model parameters and a size of the searchregion; fitting the model and the measurement data based on the selectedsample points; determining a fit level based on the fitting; evaluatingthe objective function comprising the fit level; evaluating a fitquality based on the objective function; and updating the starting pointand the search region based on the fit quality such that the objectivefunction is reduced.

According to an embodiment, the updating the starting point and thesearch region comprises selecting a new starting point and increasingthe search region, in response to the fit quality breaching a firstthreshold.

According to an embodiment, the updating the starting point and thesearch region comprises selecting a new starting point, in response tothe fit quality breaching a second threshold.

According to an embodiment, updating the starting point and the searchregion comprises decreasing a size of the search region, in response tothe fit quality breaching a third threshold.

According to an embodiment, the fitting is based on the objectivefunction comprising a cost function of second order.

According to an embodiment, the objective function includes a firstpenalty term configured to maintain a positive value of coefficients ofsecond order terms of the cost function; and/or a second penalty termassociated with a distance between predicted characteristic and themeasurement data.

Furthermore, according to an embodiment of the present disclosure, thereis provided a computer program product comprising a non-transitorycomputer readable medium having instructions recorded thereon, theinstructions when executed by a computer implementing the method of anyof the above claims.

The foregoing general description of the illustrative implementationsand the following detailed description thereof are merely exemplaryaspects of the teachings of this disclosure, and are not restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

The above aspects and other aspects and features will become apparent tothose ordinarily skilled in the art upon review of the followingdescription of specific embodiments in conjunction with the accompanyingfigures, wherein:

FIG. 1 schematically depicts a lithography apparatus, according to anembodiment;

FIG. 2 schematically depicts an embodiment of a lithographic cell orcluster, according to an embodiment;

FIG. 3 schematically depicts an example inspection apparatus andmetrology technique, according to an embodiment;

FIG. 4 schematically depicts an example inspection apparatus, accordingto an embodiment;

FIG. 5 illustrates the relationship between an illumination spot of aninspection apparatus and a metrology target, according to an embodiment;

FIG. 6 schematically depicts a process of deriving a plurality ofvariables of interest based on measurement data, according to anembodiment;

FIG. 7A is a flow chart showing various stages of a ‘design for control’process flow, according to an embodiment;

FIG. 7B is a block diagram showing various stages for visualization,according to an embodiment;

FIG. 7C is a flow chart showing how the ‘design for control’ processchooses metrology target designs robust against process perturbations,according to an embodiment;

FIG. 8 is a flow chart of a method for determining a stack configurationof a substrate, according to an embodiment;

FIG. 9A illustrates an example stack configuration, according to anembodiment;

FIG. 9B illustrates another example stack configuration, according to anembodiment;

FIG. 10 illustrates an example stack map on a substrate, according to anembodiment;

FIG. 11A illustrates a fingerprint of an example layer of a substrate,according to an embodiment;

FIG. 11B illustrates another fingerprint of an example layer of asubstrate, according to an embodiment;

FIG. 11C illustrates another fingerprint of an example layer of asubstrate, according to an embodiment;

FIG. 11D illustrates a residual fingerprint of a substrate, according toan embodiment;

FIG. 12A illustrates an example correlation between measurement andsimulated stack sensitivity at a first position on a substrate,according to an embodiment;

FIG. 12B illustrates an example correlation between measurement andsimulated stack sensitivity at a second position on a substrate,according to an embodiment;

FIG. 12C illustrates an example correlation between measurement andsimulated stack sensitivity at a third position on a substrate,according to an embodiment;

FIG. 12D illustrates an example correlation between measurement andsimulated stack sensitivity at a fourth position on a substrate,according to an embodiment;

FIG. 12E illustrates an example correlation between measurement andsimulated stack sensitivity at a fifth position on a substrate,according to an embodiment;

FIG. 13A illustrates an example thickness variation of an example layerof the substrate, according to an embodiment;

FIG. 13B illustrates an example thickness variation of another examplelayer of the substrate, according to an embodiment;

FIG. 13C illustrates an example thickness variation of yet anotherexample layer of the substrate, according to an embodiment;

FIG. 13D illustrates an example thickness variation of yet anotherexample layer of the substrate, according to an embodiment;

FIG. 14 is a method of determining optimized model parameter of a model,according to an embodiment;

FIG. 15 illustrates an example relationship to determine a sample size,according to an embodiment;

FIG. 16A illustrates an example model fitting with measurement data with30 data points for an example layer, according to an embodiment;

FIG. 16B illustrates an example model fitting with the measurement datawith 20 data points for the example layer of FIG. 16A, according to anembodiment;

FIG. 16C illustrates another example model fitting with the measurementdata with 30 data points for another example layer, according to anembodiment;

FIG. 16D illustrates another example model fitting with measurement datawith 20 data points for the example layer of FIG. 16C, according to anembodiment;

FIG. 17A illustrates an example operation of changing a search regionand center for a good fit condition based on the method of FIG. 14,according to an embodiment;

FIG. 17B illustrates an example operation of changing a search regionand center for a good fit based on the method of FIG. 14, according toan embodiment;

FIG. 17C illustrates an example convergence of the method of FIG. 14,according to an embodiment;

FIG. 18 is a block diagram of an example computer system, according toan embodiment.

FIG. 19 is a schematic diagram of a lithographic projection apparatussimilar to FIG. 1, according to an embodiment.

FIG. 20 is a schematic diagram of another lithographic projectionapparatus, according to an embodiment.

FIG. 21 is a more detailed view of the apparatus in FIG. 19, accordingto an embodiment.

FIG. 22 is a more detailed view of the source collector module SO of theapparatus of FIG. 20 and FIG. 21, according to an embodiment.

Embodiments will now be described in detail with reference to thedrawings, which are provided as illustrative examples so as to enablethose skilled in the art to practice the embodiments. Notably, thefigures and examples below are not meant to limit the scope to a singleembodiment, but other embodiments are possible by way of interchange ofsome or all of the described or illustrated elements. Whereverconvenient, the same reference numbers will be used throughout thedrawings to refer to same or like parts. Where certain elements of theseembodiments can be partially or fully implemented using knowncomponents, only those portions of such known components that arenecessary for an understanding of the embodiments will be described, anddetailed descriptions of other portions of such known components will beomitted so as not to obscure the description of the embodiments. In thepresent specification, an embodiment showing a singular component shouldnot be considered limiting; rather, the scope is intended to encompassother embodiments including a plurality of the same component, andvice-versa, unless explicitly stated otherwise herein. Moreover,applicants do not intend for any term in the specification or claims tobe ascribed an uncommon or special meaning unless explicitly set forthas such. Further, the scope encompasses present and future knownequivalents to the components referred to herein by way of illustration.

DETAILED DESCRIPTION

Although specific reference may be made in this text to the manufactureof ICs, it should be explicitly understood that the description hereinhas many other possible applications. For example, it may be employed inthe manufacture of integrated optical systems, guidance and detectionpatterns for magnetic domain memories, liquid-crystal display panels,thin-film magnetic heads, etc. The skilled artisan will appreciate that,in the context of such alternative applications, any use of the terms“reticle”, “wafer” or “die” in this text should be considered asinterchangeable with the more general terms “mask”, “substrate” and“target portion”, respectively.

FIG. 1 schematically depicts an embodiment of a lithographic apparatusLA. The apparatus comprises:

an illumination system (illuminator) IL configured to condition aradiation beam B (e.g. UV radiation or DUV radiation);

a support structure (e.g. a mask table) MT constructed to support apatterning device (e.g. a mask) MA and connected to a first positionerPM configured to accurately position the patterning device in accordancewith certain parameters;

a substrate table (e.g. a wafer table) WT (e.g., WTa, WTb or both)constructed to hold a substrate (e.g. a resist-coated wafer) W andconnected to a second positioner PW configured to accurately positionthe substrate in accordance with certain parameters; and

a projection system (e.g. a refractive projection lens system) PSconfigured to project a pattern imparted to the radiation beam B bypatterning device MA onto a target portion C (e.g. comprising one ormore dies and often referred to as fields) of the substrate W, theprojection system supported on a reference frame (RF).

As here depicted, the apparatus is of a transmissive type (e.g.employing a transmissive mask). Alternatively, the apparatus may be of areflective type (e.g. employing a programmable minor array of a type asreferred to above, or employing a reflective mask).

The illuminator IL receives a beam of radiation from a radiation sourceSO. The source and the lithographic apparatus may be separate entities,for example when the source is an excimer laser. In such cases, thesource is not considered to form part of the lithographic apparatus andthe radiation beam is passed from the source SO to the illuminator ILwith the aid of a beam delivery system BD comprising for examplesuitable directing mirrors and/or a beam expander. In other cases thesource may be an integral part of the apparatus, for example when thesource is a mercury lamp. The source SO and the illuminator IL, togetherwith the beam delivery system BD if required, may be referred to as aradiation system.

The illuminator IL may alter the intensity distribution of the beam. Theilluminator may be arranged to limit the radial extent of the radiationbeam such that the intensity distribution is non-zero within an annularregion in a pupil plane of the illuminator IL. Additionally oralternatively, the illuminator IL may be operable to limit thedistribution of the beam in the pupil plane such that the intensitydistribution is non-zero in a plurality of equally spaced sectors in thepupil plane. The intensity distribution of the radiation beam in a pupilplane of the illuminator IL may be referred to as an illumination mode.

So, the illuminator IL may comprise adjuster AM configured to adjust the(angular/spatial) intensity distribution of the beam. Generally, atleast the outer and/or inner radial extent (commonly referred to asσ-outer and σ-inner, respectively) of the intensity distribution in apupil plane of the illuminator can be adjusted. The illuminator IL maybe operable to vary the angular distribution of the beam. For example,the illuminator may be operable to alter the number, and angular extent,of sectors in the pupil plane wherein the intensity distribution isnon-zero. By adjusting the intensity distribution of the beam in thepupil plane of the illuminator, different illumination modes may beachieved. For example, by limiting the radial and angular extent of theintensity distribution in the pupil plane of the illuminator IL, theintensity distribution may have a multi-pole distribution such as, forexample, a dipole, quadrupole or hexapole distribution. A desiredillumination mode may be obtained, e.g., by inserting an optic whichprovides that illumination mode into the illuminator IL or using aspatial light modulator.

The illuminator IL may be operable alter the polarization of the beamand may be operable to adjust the polarization using adjuster AM. Thepolarization state of the radiation beam across a pupil plane of theilluminator IL may be referred to as a polarization mode. The use ofdifferent polarization modes may allow greater contrast to be achievedin the image formed on the substrate W. The radiation beam may beunpolarized. Alternatively, the illuminator may be arranged to linearlypolarize the radiation beam. The polarization direction of the radiationbeam may vary across a pupil plane of the illuminator IL. Thepolarization direction of radiation may be different in differentregions in the pupil plane of the illuminator IL. The polarization stateof the radiation may be chosen in dependence on the illumination mode.For multi-pole illumination modes, the polarization of each pole of theradiation beam may be generally perpendicular to the position vector ofthat pole in the pupil plane of the illuminator IL. For example, for adipole illumination mode, the radiation may be linearly polarized in adirection that is substantially perpendicular to a line that bisects thetwo opposing sectors of the dipole. The radiation beam may be polarizedin one of two different orthogonal directions, which may be referred toas X-polarized and Y-polarized states. For a quadrupole illuminationmode the radiation in the sector of each pole may be linearly polarizedin a direction that is substantially perpendicular to a line thatbisects that sector. This polarization mode may be referred to as XYpolarization. Similarly, for a hexapole illumination mode the radiationin the sector of each pole may be linearly polarized in a direction thatis substantially perpendicular to a line that bisects that sector. Thispolarization mode may be referred to as TE polarization.

In addition, the illuminator IL generally comprises various othercomponents, such as an integrator IN and a condenser CO. Theillumination system may include various types of optical components,such as refractive, reflective, magnetic, electromagnetic, electrostaticor other types of optical components, or any combination thereof, fordirecting, shaping, or controlling radiation.

Thus, the illuminator provides a conditioned beam of radiation B, havinga desired uniformity and intensity distribution in its cross section.

The support structure MT supports the patterning device in a manner thatdepends on the orientation of the patterning device, the design of thelithographic apparatus, and other conditions, such as for examplewhether or not the patterning device is held in a vacuum environment.The support structure can use mechanical, vacuum, electrostatic or otherclamping techniques to hold the patterning device. The support structuremay be a frame or a table, for example, which may be fixed or movable asrequired. The support structure may ensure that the patterning device isat a desired position, for example with respect to the projectionsystem. Any use of the terms “reticle” or “mask” herein may beconsidered synonymous with the more general term “patterning device.”

The term “patterning device” used herein should be broadly interpretedas referring to any device that can be used to impart a pattern in atarget portion of the substrate. In an embodiment, a patterning deviceis any device that can be used to impart a radiation beam with a patternin its cross-section so as to create a pattern in a target portion ofthe substrate. It should be noted that the pattern imparted to theradiation beam may not exactly correspond to the desired pattern in thetarget portion of the substrate, for example if the pattern includesphase-shifting features or so called assist features. Generally, thepattern imparted to the radiation beam will correspond to a particularfunctional layer in a device being created in the target portion, suchas an integrated circuit.

A patterning device may be transmissive or reflective. Examples ofpatterning devices include masks, programmable mirror arrays, andprogrammable LCD panels. Masks are well known in lithography, andinclude mask types such as binary, alternating phase-shift, andattenuated phase-shift, as well as various hybrid mask types. An exampleof a programmable mirror array employs a matrix arrangement of smallmirrors, each of which can be individually tilted so as to reflect anincoming radiation beam in different directions. The tilted mirrorsimpart a pattern in a radiation beam, which is reflected by the minormatrix.

The term “projection system” used herein should be broadly interpretedas encompassing any type of projection system, including refractive,reflective, catadioptric, magnetic, electromagnetic and electrostaticoptical systems, or any combination thereof, as appropriate for theexposure radiation being used, or for other factors such as the use ofan immersion liquid or the use of a vacuum. Any use of the term“projection lens” herein may be considered as synonymous with the moregeneral term “projection system”.

The projection system PS has an optical transfer function which may benon-uniform, which can affect the pattern imaged on the substrate W. Forunpolarized radiation such effects can be fairly well described by twoscalar maps, which describe the transmission (apodization) and relativephase (aberration) of radiation exiting the projection system PS as afunction of position in a pupil plane thereof. These scalar maps, whichmay be referred to as the transmission map and the relative phase map,may be expressed as a linear combination of a complete set of basisfunctions. A particularly convenient set is the Zernike polynomials,which form a set of orthogonal polynomials defined on a unit circle. Adetermination of each scalar map may involve determining thecoefficients in such an expansion. Since the Zernike polynomials areorthogonal on the unit circle, the Zernike coefficients may bedetermined by calculating the inner product of a measured scalar mapwith each Zernike polynomial in turn and dividing this by the square ofthe norm of that Zernike polynomial.

The transmission map and the relative phase map are field and systemdependent. That is, in general, each projection system PS will have adifferent Zernike expansion for each field point (i.e. for each spatiallocation in its image plane). The relative phase of the projectionsystem PS in its pupil plane may be determined by projecting radiation,for example from a point-like source in an object plane of theprojection system PS (i.e. the plane of the patterning device MA),through the projection system PS and using a shearing interferometer tomeasure a wavefront (i.e. a locus of points with the same phase). Ashearing interferometer is a common path interferometer and therefore,advantageously, no secondary reference beam is required to measure thewavefront. The shearing interferometer may comprise a diffractiongrating, for example a two dimensional grid, in an image plane of theprojection system (i.e. the substrate table WT) and a detector arrangedto detect an interference pattern in a plane that is conjugate to apupil plane of the projection system PS. The interference pattern isrelated to the derivative of the phase of the radiation with respect toa coordinate in the pupil plane in the shearing direction. The detectormay comprise an array of sensing elements such as, for example, chargecoupled devices (CCDs).

The projection system PS of a lithography apparatus may not producevisible fringes and therefore the accuracy of the determination of thewavefront can be enhanced using phase stepping techniques such as, forexample, moving the diffraction grating. Stepping may be performed inthe plane of the diffraction grating and in a direction perpendicular tothe scanning direction of the measurement. The stepping range may be onegrating period, and at least three (uniformly distributed) phase stepsmay be used. Thus, for example, three scanning measurements may beperformed in the y-direction, each scanning measurement being performedfor a different position in the x-direction. This stepping of thediffraction grating effectively transforms phase variations intointensity variations, allowing phase information to be determined. Thegrating may be stepped in a direction perpendicular to the diffractiongrating (z direction) to calibrate the detector.

The diffraction grating may be sequentially scanned in two perpendiculardirections, which may coincide with axes of a co-ordinate system of theprojection system PS (x and y) or may be at an angle such as 45 degreesto these axes. Scanning may be performed over an integer number ofgrating periods, for example one grating period. The scanning averagesout phase variation in one direction, allowing phase variation in theother direction to be reconstructed. This allows the wavefront to bedetermined as a function of both directions.

The transmission (apodization) of the projection system PS in its pupilplane may be determined by projecting radiation, for example from apoint-like source in an object plane of the projection system PS (i.e.the plane of the patterning device MA), through the projection system PSand measuring the intensity of radiation in a plane that is conjugate toa pupil plane of the projection system PS, using a detector. The samedetector as is used to measure the wavefront to determine aberrationsmay be used.

The projection system PS may comprise a plurality of optical (e.g.,lens) elements and may further comprise an adjustment mechanism AMconfigured to adjust one or more of the optical elements so as tocorrect for aberrations (phase variations across the pupil planethroughout the field). To achieve this, the adjustment mechanism may beoperable to manipulate one or more optical (e.g., lens) elements withinthe projection system PS in one or more different ways. The projectionsystem may have a co-ordinate system wherein its optical axis extends inthe z direction. The adjustment mechanism may be operable to do anycombination of the following: displace one or more optical elements;tilt one or more optical elements; and/or deform one or more opticalelements. Displacement of an optical element may be in any direction (x,y, z or a combination thereof). Tilting of an optical element istypically out of a plane perpendicular to the optical axis, by rotatingabout an axis in the x and/or y directions although a rotation about thez axis may be used for a non-rotationally symmetric aspherical opticalelement. Deformation of an optical element may include a low frequencyshape (e.g. astigmatic) and/or a high frequency shape (e.g. free formaspheres). Deformation of an optical element may be performed forexample by using one or more actuators to exert force on one or moresides of the optical element and/or by using one or more heatingelements to heat one or more selected regions of the optical element. Ingeneral, it may not be possible to adjust the projection system PS tocorrect for apodization (transmission variation across the pupil plane).The transmission map of a projection system PS may be used whendesigning a patterning device (e.g., mask) MA for the lithographyapparatus LA. Using a computational lithography technique, thepatterning device MA may be designed to at least partially correct forapodization.

The lithographic apparatus may be of a type having two (dual stage) ormore tables (e.g., two or more substrate tables WTa, WTb, two or morepatterning device tables, a substrate table WTa and a table WTb belowthe projection system without a substrate that is dedicated to, forexample, facilitating measurement, and/or cleaning, etc.). In such“multiple stage” machines the additional tables may be used in parallel,or preparatory steps may be carried out on one or more tables while oneor more other tables are being used for exposure. For example, alignmentmeasurements using an alignment sensor AS and/or level (height, tilt,etc.) measurements using a level sensor LS may be made.

The lithographic apparatus may also be of a type wherein at least aportion of the substrate may be covered by a liquid having a relativelyhigh refractive index, e.g. water, so as to fill a space between theprojection system and the substrate. An immersion liquid may also beapplied to other spaces in the lithographic apparatus, for example,between the patterning device and the projection system. Immersiontechniques are well known in the art for increasing the numericalaperture of projection systems. The term “immersion” as used herein doesnot mean that a structure, such as a substrate, must be submerged inliquid, but rather only means that liquid is located between theprojection system and the substrate during exposure.

So, in operation of the lithographic apparatus, a radiation beam isconditioned and provided by the illumination system IL. The radiationbeam B is incident on the patterning device (e.g., mask) MA, which isheld on the support structure (e.g., mask table) MT, and is patterned bythe patterning device. Having traversed the patterning device MA, theradiation beam B passes through the projection system PS, which focusesthe beam onto a target portion C of the substrate W. With the aid of thesecond positioner PW and position sensor IF (e.g. an interferometricdevice, linear encoder, 2-D encoder or capacitive sensor), the substratetable WT can be moved accurately, e.g. so as to position differenttarget portions C in the path of the radiation beam B. Similarly, thefirst positioner PM and another position sensor (which is not explicitlydepicted in FIG. 1) can be used to accurately position the patterningdevice MA with respect to the path of the radiation beam B, e.g. aftermechanical retrieval from a mask library, or during a scan. In general,movement of the support structure MT may be realized with the aid of along-stroke module (coarse positioning) and a short-stroke module (finepositioning), which form part of the first positioner PM. Similarly,movement of the substrate table WT may be realized using a long-strokemodule and a short-stroke module, which form part of the secondpositioner PW. In the case of a stepper (as opposed to a scanner) thesupport structure MT may be connected to a short-stroke actuator only,or may be fixed. Patterning device MA and substrate W may be alignedusing patterning device alignment marks M1, M2 and substrate alignmentmarks P1, P2. Although the substrate alignment marks as illustratedoccupy dedicated target portions, they may be located in spaces betweentarget portions (these are known as scribe-lane alignment marks).Similarly, in situations in which more than one die is provided on thepatterning device MA, the patterning device alignment marks may belocated between the dies.

The depicted apparatus could be used in at least one of the followingmodes:

1. In step mode, the support structure MT and the substrate table WT arekept essentially stationary, while an entire pattern imparted to theradiation beam is projected onto a target portion C at one time (i.e. asingle static exposure). The substrate table WT is then shifted in the Xand/or Y direction so that a different target portion C can be exposed.In step mode, the maximum size of the exposure field limits the size ofthe target portion C imaged in a single static exposure.

2. In scan mode, the support structure MT and the substrate table WT arescanned synchronously while a pattern imparted to the radiation beam isprojected onto a target portion C (i.e. a single dynamic exposure). Thevelocity and direction of the substrate table WT relative to the supportstructure MT may be determined by the (de-)magnification and imagereversal characteristics of the projection system PS. In scan mode, themaximum size of the exposure field limits the width (in the non-scanningdirection) of the target portion in a single dynamic exposure, whereasthe length of the scanning motion determines the height (in the scanningdirection) of the target portion.

3. In another mode, the support structure MT is kept essentiallystationary holding a programmable patterning device, and the substratetable WT is moved or scanned while a pattern imparted to the radiationbeam is projected onto a target portion C. In this mode, generally apulsed radiation source is employed and the programmable patterningdevice is updated as required after each movement of the substrate tableWT or in between successive radiation pulses during a scan. This mode ofoperation can be readily applied to maskless lithography that utilizesprogrammable patterning device, such as a programmable mirror array of atype as referred to above.

Combinations and/or variations on the above described modes of use orentirely different modes of use may also be employed.

Although specific reference may be made in this text to the use oflithography apparatus in the manufacture of ICs, it should be understoodthat the lithography apparatus described herein may have otherapplications, such as the manufacture of integrated optical systems,guidance and detection patterns for magnetic domain memories,liquid-crystal displays (LCDs), thin film magnetic heads, etc. Theskilled artisan will appreciate that, in the context of such alternativeapplications, any use of the terms “wafer” or “die” herein may beconsidered as synonymous with the more general terms “substrate” or“target portion”, respectively. The substrate referred to herein may beprocessed, before or after exposure, in for example a track (a tool thattypically applies a layer of resist to a substrate and develops theexposed resist) or a metrology or inspection tool. Where applicable, thedisclosure herein may be applied to such and other substrate processingtools. Further, the substrate may be processed more than once, forexample in order to create a multi-layer IC, so that the term substrateused herein may also refer to a substrate that already contains multipleprocessed layers.

The terms “radiation” and “beam” used herein encompass all types ofelectromagnetic radiation, including ultraviolet (UV) radiation (e.g.having a wavelength of 365, 248, 193, 157 or 126 nm) and extremeultra-violet (EUV) radiation (e.g. having a wavelength in the range of5-20 nm), as well as particle beams, such as ion beams or electronbeams.

Various patterns on or provided by a patterning device may havedifferent process windows. i.e., a space of processing variables underwhich a pattern will be produced within specification. Examples ofpattern specifications that relate to potential systematic defectsinclude checks for necking, line pull back, line thinning, CD, edgeplacement, overlapping, resist top loss, resist undercut and/orbridging. The process window of all the patterns on a patterning deviceor an area thereof may be obtained by merging (e.g., overlapping)process windows of each individual pattern. The boundary of the processwindow of all the patterns contains boundaries of process windows ofsome of the individual patterns. In other words, these individualpatterns limit the process window of all the patterns. These patternscan be referred to as “hot spots” or “process window limiting patterns(PWLPs),” which are used interchangeably herein. When controlling a partof a patterning process, it is possible and economical to focus on thehot spots. When the hot spots are not defective, it is most likely thatall the patterns are not defective.

As shown in FIG. 2, the lithographic apparatus LA may form part of alithographic cell LC, also sometimes referred to a lithocell or cluster,which also includes apparatuses to perform pre- and post-exposureprocesses on a substrate. Conventionally these include one or more spincoaters SC to deposit one or more resist layers, one or more developersDE to develop exposed resist, one or more chill plates CH and/or one ormore bake plates BK. A substrate handler, or robot, RO picks up one ormore substrates from input/output port I/O1, I/O2, moves them betweenthe different process apparatuses and delivers them to the loading bayLB of the lithographic apparatus. These apparatuses, which are oftencollectively referred to as the track, are under the control of a trackcontrol unit TCU which is itself controlled by the supervisory controlsystem SCS, which also controls the lithographic apparatus vialithography control unit LACU. Thus, the different apparatuses can beoperated to maximize throughput and processing efficiency.

In order that a substrate that is exposed by the lithographic apparatusis exposed correctly and consistently and/or in order to monitor a partof the patterning process (e.g., a device manufacturing process) thatincludes at least one pattern transfer step (e.g., an opticallithography step), it is desirable to inspect a substrate or otherobject to measure or determine one or more properties such as alignment,overlay (which can be, for example, between structures in overlyinglayers or between structures in a same layer that have been providedseparately to the layer by, for example, a double patterning process),line thickness, critical dimension (CD), focus offset, a materialproperty, etc. Accordingly a manufacturing facility in which lithocellLC is located also typically includes a metrology system MET whichmeasures some or all of the substrates W that have been processed in thelithocell or other objects in the lithocell. The metrology system METmay be part of the lithocell LC, for example it may be part of thelithographic apparatus LA (such as alignment sensor AS).

The one or more measured parameters may include, for example, overlaybetween successive layers formed in or on the patterned substrate,critical dimension (CD) (e.g., critical linewidth) of, for example,features formed in or on the patterned substrate, focus or focus errorof an optical lithography step, dose or dose error of an opticallithography step, optical aberrations of an optical lithography step,etc. This measurement may be performed on a target of the productsubstrate itself and/or on a dedicated metrology target provided on thesubstrate. The measurement can be performed after-development of aresist but before etching or can be performed after-etch.

There are various techniques for making measurements of the structuresformed in the patterning process, including the use of a scanningelectron microscope, an image-based measurement tool and/or variousspecialized tools. As discussed above, a fast and non-invasive form ofspecialized metrology tool is one in which a beam of radiation isdirected onto a target on the surface of the substrate and properties ofthe scattered (diffracted/reflected) beam are measured. By evaluatingone or more properties of the radiation scattered by the substrate, oneor more properties of the substrate can be determined. This may betermed diffraction-based metrology. One such application of thisdiffraction-based metrology is in the measurement of feature asymmetrywithin a target. This can be used as a measure of overlay, for example,but other applications are also known. For example, asymmetry can bemeasured by comparing opposite parts of the diffraction spectrum (forexample, comparing the −1st and +1st orders in the diffraction spectrumof a periodic grating). This can be done as described above and asdescribed, for example, in U.S. patent application publication US2006-066855, which is incorporated herein in its entirety by reference.Another application of diffraction-based metrology is in the measurementof feature width (CD) within a target. Such techniques can use theapparatus and methods described hereafter.

Thus, in a device fabrication process (e.g., a patterning process or alithography process), a substrate or other objects may be subjected tovarious types of measurement during or after the process. Themeasurement may determine whether a particular substrate is defective,may establish adjustments to the process and apparatuses used in theprocess (e.g., aligning two layers on the substrate or aligning thepatterning device to the substrate), may measure the performance of theprocess and the apparatuses, or may be for other purposes. Examples ofmeasurement include optical imaging (e.g., optical microscope),non-imaging optical measurement (e.g., measurement based on diffractionsuch as ASML YieldStar metrology tool, ASML SMASH metrology system),mechanical measurement (e.g., profiling using a stylus, atomic forcemicroscopy (AFM)), and/or non-optical imaging (e.g., scanning electronmicroscopy (SEM)). The SMASH (SMart Alignment Sensor Hybrid) system, asdescribed in U.S. Pat. No. 6,961,116, which is incorporated by referenceherein in its entirety, employs a self-referencing interferometer thatproduces two overlapping and relatively rotated images of an alignmentmarker, detects intensities in a pupil plane where Fourier transforms ofthe images are caused to interfere, and extracts the positionalinformation from the phase difference between diffraction orders of thetwo images which manifests as intensity variations in the interferedorders.

Metrology results may be provided directly or indirectly to thesupervisory control system SCS. If an error is detected, an adjustmentmay be made to exposure of a subsequent substrate (especially if theinspection can be done soon and fast enough that one or more othersubstrates of the batch are still to be exposed) and/or to subsequentexposure of the exposed substrate. Also, an already exposed substratemay be stripped and reworked to improve yield, or discarded, therebyavoiding performing further processing on a substrate known to befaulty. In a case where only some target portions of a substrate arefaulty, further exposures may be performed only on those target portionswhich are good.

Within a metrology system MET, a metrology apparatus is used todetermine one or more properties of the substrate, and in particular,how one or more properties of different substrates vary or differentlayers of the same substrate vary from layer to layer. As noted above,the metrology apparatus may be integrated into the lithographicapparatus LA or the lithocell LC or may be a stand-alone device.

To enable the metrology, one or more targets can be provided on thesubstrate. In an embodiment, the target is specially designed and maycomprise a periodic structure. In an embodiment, the target is a part ofa device pattern, e.g., a periodic structure of the device pattern. Inan embodiment, the device pattern is a periodic structure of a memorydevice (e.g., a Bipolar Transistor (BPT), a Bit Line Contact (BLC), etc.structure).

In an embodiment, the target on a substrate may comprise one or more 1-Dperiodic structures (e.g., gratings), which are printed such that afterdevelopment, the periodic structural features are formed of solid resistlines. In an embodiment, the target may comprise one or more 2-Dperiodic structures (e.g., gratings), which are printed such that afterdevelopment, the one or more periodic structures are formed of solidresist pillars or vias in the resist. The bars, pillars or vias mayalternatively be etched into the substrate (e.g., into one or morelayers on the substrate).

In an embodiment, one of the parameters of interest of a patterningprocess is overlay. Overlay can be measured using dark fieldscatterometry in which the zeroth order of diffraction (corresponding toa specular reflection) is blocked, and only higher orders processed.Examples of dark field metrology can be found in PCT patent applicationpublication nos. WO 2009/078708 and WO 2009/106279, which are herebyincorporated in their entirety by reference. Further developments of thetechnique have been described in U.S. patent application publicationsUS2011-0027704, US2011-0043791 and US2012-0242970, which are herebyincorporated in their entirety by reference. Diffraction-based overlayusing dark-field detection of the diffraction orders enables overlaymeasurements on smaller targets. These targets can be smaller than theillumination spot and may be surrounded by device product structures ona substrate. In an embodiment, multiple targets can be measured in oneradiation capture.

FIG. 3 depicts an example inspection apparatus (e.g., a scatterometer).It comprises a broadband (white light) radiation projector 2 whichprojects radiation onto a substrate W. The redirected radiation ispassed to a spectrometer detector 4, which measures a spectrum 10(intensity as a function of wavelength) of the specular reflectedradiation, as shown, e.g., in the graph in the lower left of FIG. 3.From this data, the structure or profile giving rise to the detectedspectrum may be reconstructed by processor PU, e.g. by Rigorous CoupledWave Analysis and non-linear regression or by comparison with a libraryof simulated spectra as shown at the bottom right of FIG. 3. In general,for the reconstruction the general form of the structure is known andsome variables are assumed from knowledge of the process by which thestructure was made, leaving only a few variables of the structure to bedetermined from the measured data. Such an inspection apparatus may beconfigured as a normal-incidence inspection apparatus or anoblique-incidence inspection apparatus.

Another inspection apparatus that may be used is shown in FIG. 4. Inthis device, the radiation emitted by radiation source 2 is collimatedusing lens system 12 and transmitted through interference filter 13 andpolarizer 17, reflected by partially reflecting surface 16 and isfocused into a spot S on substrate W via an objective lens 15, which hasa high numerical aperture (NA), desirably at least 0.9 or at least 0.95.An immersion inspection apparatus (using a relatively high refractiveindex fluid such as water) may even have a numerical aperture over 1.

As in the lithographic apparatus LA, one or more substrate tables may beprovided to hold the substrate W during measurement operations. Thesubstrate tables may be similar or identical in form to the substratetable WT of FIG. 1. In an example where the inspection apparatus isintegrated with the lithographic apparatus, they may even be the samesubstrate table. Coarse and fine positioners may be provided to a secondpositioner PW configured to accurately position the substrate inrelation to a measurement optical system. Various sensors and actuatorsare provided for example to acquire the position of a target ofinterest, and to bring it into position under the objective lens 15.Typically many measurements will be made on targets at differentlocations across the substrate W. The substrate support can be moved inX and Y directions to acquire different targets, and in the Z directionto obtain a desired location of the target relative to the focus of theoptical system. It is convenient to think and describe operations as ifthe objective lens is being brought to different locations relative tothe substrate, when, for example, in practice the optical system mayremain substantially stationary (typically in the X and Y directions,but perhaps also in the Z direction) and only the substrate moves.Provided the relative position of the substrate and the optical systemis correct, it does not matter in principle which one of those is movingin the real world, or if both are moving, or a combination of a part ofthe optical system is moving (e.g., in the Z and/or tilt direction) withthe remainder of the optical system being stationary and the substrateis moving (e.g., in the X and Y directions, but also optionally in the Zand/or tilt direction).

The radiation redirected by the substrate W then passes throughpartially reflecting surface 16 into a detector 18 in order to have thespectrum detected. The detector 18 may be located at a back-projectedfocal plane 11 (i.e., at the focal length of the lens system 15) or theplane 11 may be re-imaged with auxiliary optics (not shown) onto thedetector 18. The detector may be a two-dimensional detector so that atwo-dimensional angular scatter spectrum of a substrate target 30 can bemeasured. The detector 18 may be, for example, an array of CCD or CMOSsensors, and may use an integration time of, for example, 40milliseconds per frame.

A reference beam may be used, for example, to measure the intensity ofthe incident radiation. To do this, when the radiation beam is incidenton the partially reflecting surface 16 part of it is transmitted throughthe partially reflecting surface 16 as a reference beam towards areference mirror 14. The reference beam is then projected onto adifferent part of the same detector 18 or alternatively on to adifferent detector (not shown).

One or more interference filters 13 are available to select a wavelengthof interest in the range of, say, 405-790 nm or even lower, such as200-300 nm. The interference filter may be tunable rather thancomprising a set of different filters. A grating could be used insteadof an interference filter. An aperture stop or spatial light modulator(not shown) may be provided in the illumination path to control therange of angle of incidence of radiation on the target.

The detector 18 may measure the intensity of redirected radiation at asingle wavelength (or narrow wavelength range), the intensity separatelyat multiple wavelengths or integrated over a wavelength range.Furthermore, the detector may separately measure the intensity oftransverse magnetic- and transverse electric-polarized radiation and/orthe phase difference between the transverse magnetic- and transverseelectric-polarized radiation.

The target 30 on substrate W may be a 1-D grating, which is printed suchthat after development, the bars are formed of solid resist lines. Thetarget 30 may be a 2-D grating, which is printed such that afterdevelopment, the grating is formed of solid resist pillars or vias inthe resist. The bars, pillars or vias may be etched into or on thesubstrate (e.g., into one or more layers on the substrate). The pattern(e.g., of bars, pillars or vias) is sensitive to change in processing inthe patterning process (e.g., optical aberration in the lithographicprojection apparatus (particularly the projection system PS), focuschange, dose change, etc.) and will manifest in a variation in theprinted grating. Accordingly, the measured data of the printed gratingis used to reconstruct the grating. One or more parameters of the 1-Dgrating, such as line width and/or shape, or one or more parameters ofthe 2-D grating, such as pillar or via width or length or shape, may beinput to the reconstruction process, performed by processor PU, fromknowledge of the printing step and/or other inspection processes.

In addition to measurement of a parameter by reconstruction, angleresolved scatterometry is useful in the measurement of asymmetry offeatures in product and/or resist patterns. A particular application ofasymmetry measurement is for the measurement of overlay, where thetarget 30 comprises one set of periodic features superimposed onanother. The concepts of asymmetry measurement using the instrument ofFIG. 3 or FIG. 4 are described, for example, in U.S. patent applicationpublication US2006-066855, which is incorporated herein in its entirety.Simply stated, while the positions of the diffraction orders in thediffraction spectrum of the target are determined only by theperiodicity of the target, asymmetry in the diffraction spectrum isindicative of asymmetry in the individual features which make up thetarget. In the instrument of FIG. 4, where detector 18 may be an imagesensor, such asymmetry in the diffraction orders appears directly asasymmetry in the pupil image recorded by detector 18. This asymmetry canbe measured by digital image processing in unit PU, and calibratedagainst known values of overlay.

FIG. 5 illustrates a plan view of a typical target 30, and the extent ofillumination spot S in the apparatus of FIG. 4. To obtain a diffractionspectrum that is free of interference from surrounding structures, thetarget 30, in an embodiment, is a periodic structure (e.g., grating)larger than the width (e.g., diameter) of the illumination spot S. Thewidth of spot S may be smaller than the width and length of the target.The target in other words is ‘underfilled’ by the illumination, and thediffraction signal is essentially free from any signals from productfeatures and the like outside the target itself. The illuminationarrangement 2, 12, 13, 17 may be configured to provide illumination of auniform intensity across a back focal plane of objective 15.Alternatively, by, e.g., including an aperture in the illumination path,illumination may be restricted to on axis or off axis directions.

FIG. 6 schematically depicts an example process of the determination ofthe value of one or more variables of interest of a target pattern 30′based on measurement data obtained using metrology. Radiation detectedby the detector 18 provides a measured radiation distribution 608 fortarget 30′.

For a given target 30′, a radiation distribution 612 can becomputed/simulated from a parameterized model 606 using, for example, anumerical Maxwell solver 610. The parameterized model 606 shows examplelayers of various materials making up, and associated with, the target.The parameterized model 606 may include one or more of variables for thefeatures and layers of the portion of the target under consideration,which may be varied and derived. As shown in FIG. 6, the one or more ofthe variables may include the thickness t of one or more layers, a widthw (e.g., CD) of one or more features, a height h of one or morefeatures, and/or a sidewall angle α of one or more features. Althoughnot shown, the one or more of the variables may further include, but isnot limited to, the refractive index (e.g., a real or complex refractiveindex, refractive index tensor, etc.) of one or more of the layers, theextinction coefficient of one or more layers, the absorption of one ormore layers, resist loss during development, a footing of one or morefeatures, and/or line edge roughness of one or more features. Theinitial values of the variables may be those expected for the targetbeing measured. The measured radiation distribution 608 is then comparedat 612 to the computed radiation distribution 612 to determine thedifference between the two. If there is a difference, the values of oneor more of the variables of the parameterized model 606 may be varied, anew computed radiation distribution 612 calculated and compared againstthe measured radiation distribution 608 until there is sufficient matchbetween the measured radiation distribution 608 and the computedradiation distribution 612. At that point, the values of the variablesof the parameterized model 606 provide a good or best match of thegeometry of the actual target 30′. In an embodiment, there is sufficientmatch when a difference between the measured radiation distribution 608and the computed radiation distribution 612 is within a tolerancethreshold.

FIG. 7A shows a flowchart that lists the main stages of the D4C method.In stage 710, the materials to be used in the lithography process areselected. The materials may be selected from a materials libraryinterfaced with D4C through appropriate GUI. In stage 720, a lithographyprocess is defined by entering each of the process steps, and building acomputer simulation model for the entire process sequence. In stage 730,a metrology target is defined, i.e. dimensions and other characteristicsof various features included in the target are entered into the D4Cprogram. For example, if a grating is included in a structure, thennumber of grating elements, width of individual grating elements,spacing between two grating elements etc. have to be defined. In stage740, the 3D geometry is created. This step also takes into account ifthere is any information relevant to a multi-layer target design, forexample, the relative shifts between different layers. This featureenables multi-layer target design. In stage 750, the final geometry ofthe designed target is visualized. As will be explained in greaterdetail below, not only the final design is visualized, but as thedesigner applies various steps of the lithography process, he/she canvisualize how the 3D geometry is being formed and changed because ofprocess-induced effects. For example, the 3D geometry after resistpatterning is different from the 3D geometry after resist removal andetching.

An important aspect of the present disclosure is that the targetdesigner is enabled to visualize the stages of the method to facilitatetheir perception and control during modeling and simulation. Differentvisualization tools, referred to as “viewers,” are built into the D4Csoftware. For example, as shown in FIG. 7B, a designer can view materialplots 760 (and may also get a run time estimation plot) depending on thedefined lithography process and target. Once the lithography model iscreated, the designer can view the model parameters through model viewertool 775. Design layout viewer tool 780 may be used to view the designlayout (e.g., visual rendering of the GDS file). Resist profile viewertool 785 may be used to view pattern profiles in a resist. Geometryviewer tool 790 may be used to view 3D structures on a substrate. Apupil viewer tool 795 may be used to view simulated response on ametrology tool. Persons skilled in the art would understand that theseviewing tools are available to enhance the understanding of the designerduring design and simulation. One or more of these tools may not bepresent in some embodiments of D4C software, and additional viewingtools may be there in some other embodiments.

FIG. 7C shows a flow chart that illustrates how the D4C processincreases efficiency in the overall simulation process by reducing thenumber of metrology targets selected for the actual simulation of thelithography process. As mentioned before, D4C enables designers todesign thousands or even millions of designs. Not all of these designsmay be robust against variations in the process steps. To select asubset of target designs that can withstand process variation, alithographer may intentionally perturb one or more steps of the definedlithography process, as shown in block 752. The introduction of theperturbation alters the entire process sequence with respect to how itwas originally defined. Therefore, applying the perturbed processsequence (block 754) alters the 3D geometry of the designed target too.A lithographer only selects the perturbations that show nonzeroalternations in the original design targets and creates a subset ofselected process perturbations (block 756). The lithography process isthen simulated with this subset of process perturbations (block 758).

The manufacturing or fabrication of a substrate using the lithographicprocess (or patterning process in general) typically involves processvariations. The process variations are not uniform across the substrate.For example, in deposition process, films tend to be thicker at thecenter of the substrate and be thinner when close to edge. Thesesystematic variations are usually reflected in measurements data as‘fingerprints’, which are characteristics of a substrate based on knownprocess conditions. In other words, there exists a stack on a substratethat has a spatial variation as a function of substrate coordinate. Astack comprises multiple layers formed on a substrate during thepatterning process to form a selected pattern (e.g., a design pattern)on the substrate. Each layer of the stack can be associated with athickness, material properties, and features and related parameters ofthe patterning process (e.g. CD, pitch, overlay, etc.).

According to the present disclosure, the stack is modeled to predict astack configuration based on the location on a substrate such that thepredicted stack configuration matches the measurement data of the stackconfiguration. The process of building the model and predicting thestack configuration using the model at a particular location is alsoreferred as stack tuning or stack reconstruction. In other words, modelparameters of the model are modified or tuned till an optimum stackconfiguration is generated. The existing strategy of stack tuning isbased on using all measurements to tune one single stack, however suchstack tuning does not match measurements in cases that involvesubstantial process variations at different locations across thesubstrate. Furthermore, conventionally, a stack tuning tool involves afield engineer or a computer scientist manually tuning an inaccuratestack by trial and error to match the metrology measurements. This is amanually intensive and error prone process that usually takessubstantial amount of time but the outcomes are often not satisfying.

Stack-tuning/stack reconstruction/stack configuration is a challengingand demanding task in metrology applications (e.g., using scatterometer,or a Yield Star metrology tool). There are many factors that contributeto the non-ideal correlations between the metrology measurement and D4Csimulation. These factors include, but are not limited to, inaccurateprocess stack information, inaccurate materials n, k information, systemnoise, process variations, etc. These factors make the interpretation ofthe measurement data and generating the metrology target design during asecond time (e.g., in a subsequent substrate processing) a challengingtask.

A reconstructed stack, which yields good correlations betweenmeasurement and simulation by considering the slight deviation ofparameters of the patterning process from a selected value (e.g.,nominal values related to CD, pitch, etc.), is highly desired to achievesimulation accuracy, expedited second time target design and yieldimprovement.

FIG. 8 is a method for determining a stack configuration at a particularlocation of a substrate subjected a patterning process. The methodenables defining an optimum stack configuration by considering processvariations across the substrate. The optimum stack configuration isbased on a stack model that accounts for locations on a substrate anddetermines model parameters of the stack model in an iterative manner.

The terms “optimizing” and “optimization” as used herein refer to ormean adjusting values of the model parameters of the model of a stackcharacteristic that is further used to define the stack configuration.In an embodiment, adjusting may be of an apparatus and/or process of thepatterning process, which may include adjusting a lithography process orapparatus, or adjusting the metrology process or apparatus (e.g., thetarget, measurement tool, etc.), such that a figure of merit has a moredesirable value, such as patterning and/or device fabrication resultsand/or processes (e.g., of lithography) have one or more desirablecharacteristics, projection of a design layout on a substrate being moreaccurate, a process window being larger, etc. Thus, optimizing andoptimization can also refer to or mean a process that identifies one ormore values for one or more design variables (e.g., stackcharacteristics or a corresponding stack configuration) that provide animprovement, e.g., a local optimum, in a figure of merit, compared to aninitial set of values of the design variables. “Optimum” and otherrelated terms should be construed accordingly. In an embodiment,optimization steps can be applied iteratively to provide furtherimprovements in one or more figures of merit.

The method, in process P801, includes obtaining (i) measurement data 801of a stack configuration with location information of a stackcharacteristic on a printed substrate, and (ii) a substrate model 803configured to predict a stack characteristic based on a location of thesubstrate. In an embodiment, user inputs 802 (e.g., initial values ofmodel parameters of the substrate model) may also be obtained toinitiate the substrate model 803. In an embodiment, a substrate map maybe obtained and/or generated (as discussed in process P804).

A stack configuration refers to an arrangement of different layersrelative to each other that may be formed on the substrate during thepatterning process. In an embodiment, the stack configuration includes aplurality of layers and information related to each layer. For example,each layer may be associated with a geometry, a material, or otherinformation. In an embodiment, each layer is associated with a layerthickness, one or more feature on the layer, a location of the stack onthe substrate, and/or material information (e.g., n, k values) of thelayer, relative position of the layer with respect to other layers, etc.

The stack configuration may be defined in terms of one or more stackcharacteristics. The stack characteristic may be a parameter of thesubstrate related to the feature, the geometry, or the material of thesubstrate. In an embodiment, the stack characteristic may be a thicknessof a layer of the substrate, a critical dimension of a feature of thesubstrate, and/or a distance between adjacent features of the substrate.In an embodiment, the stack characteristic is a difference in thicknessof the layer and a selected thickness of the layer (e.g., a nominalthickness of the layer, an average thickness of the layer, or anintended design thickness). In an embodiment, the stack characteristiccan be a measured, simulated, and/or a derived parameter.

FIGS. 9A and 9B illustrate example stack configurations. FIG. 9A is across-section of an example stack configuration 900 including differentlayers, features, etc. and FIG. 9B is a three dimensional representationof a stack configuration 920. In FIG. 9A, the stack configuration 900includes layers 902, 904, 906, 908, 910, etc. formed during the varioussteps of the patterning process. For example, the layer 902 may be aresist layer (or an etch layer), the layer 904 may be an oxide layermade of a first oxide (e.g., SOH based) formed by deposition, the layer906 may be a second oxide layer (e.g., amorphous silicon oxide based),the layer 910 may be a etch layer, etc. Furthermore, each layer hasdifferent properties such as a material property, geometric propertysuch as a thickness, which can be measured, for example, using ametrology tool as discussed earlier, etc. In an addition, each layer mayinclude one or more features having characteristics such as a CD, pitch,etc., which may also be measured using the metrology tool. Themeasurements from the metrology tool may be included in one or moreitems of measurement data.

In an embodiment, as shown in FIG. 9B, a more complex device structuressuch as a FinFET array may be determined that requires several layers ofmaterials and several process steps. In such structures, overlay controlmay be the parameter of interest that may be optimized during the stackreconfiguration. For example, a top electrode layer 922 is aligned withrespect to the array of fins 924 in a layer which is not necessarilyadjacent to the top electrode layer. Stack reconfiguration using D4Csimulation allows to define spatial and other characteristicrelationship between features from different layers.

The measurement data 801 is related to one or more stack characteristicsof the printed substrate that may be obtained from metrology tools(e.g., scatterometer, interferometer, etc.), as discussed earlier in thepresent disclosure. In an embodiment, the measurement data 801 includesinformation related to the stack configuration at a particular locationon the substrate. For example, measurement data 801 can include aplurality of stack characteristics such as a thickness information of afirst layer (e.g., an etch layer), a second layer (e.g., an amorphoussilicon oxide layer), a third layer (spin-on hard mask layer), a fourthlayer (e.g., photo resist layer), and so on at each of the locations P1,P2, P3, P4, P5, etc. These locations are spread across the substrate.The locations may be identified in the form of Cartesian coordinates(x,y) or polar coordinates (r, θ). In an embodiment, the measurementdata may be converted from a Cartesian coordinates to polar coordinatesusing, for example, geometric correlation between Cartesian to polarconversions, and/or using a Zernike based conversion model. The Zernikebased conversion model uses Zernike polynomials which enables capture ofvariation in thickness across the substrate as such variation tends tobe radially symmetric while Zernike polynomials are well-suited todescribing radially symmetric systems.

Additionally, a metrology recipe data such as a setting of the metrologytool (e.g., Yieldstar, scatterometer, etc.) may be obtained or includedin the measurement data 801. The recipe data includes, for example,wavelength, polarization, light source intensity, etc. The recipe datacan also be associated with a location on the substrate. The recipe canbe associated with a stack characteristic of the stack configuration ata particular location. Thus, appropriate recipe may be selected duringthe metrology to obtain accurate measurements of a metrology target (orfeatures) on the substrate.

The substrate model 803 is a mathematical model related to a stackcharacteristic (e.g., overlay, thickness, sidewall angle, etc.) definedin terms of a location on a substrate. The substrate model 803 includesa plurality of model parameters or tuning parameters, which can be tunedbased on the measurement data, according the method of presentdisclosure. In an embodiment, the substrate model 803 may be astatistical model, for example, a linear regression model, a secondorder (e.g., having second order terms) model, or other higher orderregression model. In an embodiment, the substrate model 803 may be acollection of mathematical models defined for each of the stackcharacteristics.

In an embodiment, the substrate model 803 (e.g., represented byequations 1 and 2) may be a thickness based model in Cartesiancoordinates having a first set of model parameters (also referred astuning parameters) and a second set of model parameters, defined asfollows:

Δt _(A) =k _(A1) +k _(A7)*(x ² +y ²)   (1)

Δt _(B) =k _(B1) +k _(B7)*(x ² +y ²)   (2)

In the above equations 1 and 2, Δt_(A) is a difference in thickness of afirst layer (layer A) and a nominal thickness of the first layer, Δt_(B)is a difference in thickness of a second layer (layer B) and a nominalthickness of the second layer, k_(A1) and k_(A7) are tuning parametersof the first substrate model, k_(B1) and k_(B7) are tuning parameters ofthe second substrate model, and x and y are the Cartesian coordinatescorresponding to each of the locations such as P1, P2, P3, P4, P5, etc.across the substrate. Thus, the location-specific value of the stackcharacteristic may be determined. The substrate model 803 may beevaluated for all locations on a substrate and the tuning parameters maybe determined based on the measurement data 801 an objective function.After the tuning process or optimization process, the tuning parameterswill have specific values based on which a thickness of a layer at anylocation may be determined with high accuracy, as such improvingaccuracy of the metrology tool.

In an embodiment, the substrate model 803 may be based on an overlay, analignment and/or a sidewall angle of features of the substrate.Furthermore, the substrate model 803 may include different terms andassociated model parameters. Each term may be associated with an aspectof the patterning process. Accordingly, the substrate model 803 mayinclude terms such as a substrate leveling term to mimic a translationoperation and/or rotation (e.g., in x or y direction), magnificationterms (e.g., in x or y direction) corresponding to a lens of the opticalsystem, scanning directions/patterns (e.g., in x or y direction) termsusing a lens, bow factors terms of the lens, a third order magnificationfactor related terms, C-shape distortion terms, etc. Each term may bedescribed in terms of x-y location on a substrate. Further each term maybe associated (e.g., multiplied) with a model parameter (e.g., k1, k2,k3, k4, k5, k6, etc.). At the end of the iterations of the presentmethod, optimized values of the parameters k1, k2, k3, k4, etc. areobtained that enables accurate prediction of the stack characteristic(e.g., thickness, overlay, alignment) and the stack configuration.

In an embodiment, the substrate model 803, which may be a radius basedmodel (e.g., represented by equations 3 and 4) may be expressed in polarcoordinates and model parameters as Zernike polynomials. These modelparameters are referred as a second set of model parameters, as shown inequations 2-4 below. Thus, the second set of model parameters capturesprocess variations in the form of Zernike polynomials. Such substratemodel 803 may be applied to account for process variations due to, forexample, deposition, etching, CMP, etc. that leave a strong radialfingerprint (e.g., variation in thickness, overlay or alignmentcharacteristics across the entire substrate). An example, thicknessbased substrate model 803 is described as follows:

Δt _(A) =Z _(A1) +Z _(A4) ×R ²   (3)

Δt _(B) =Z _(B1) +Z _(B4) ×R ²   (4)

In the above equations 3 and 4, Δt_(A) is a difference in thickness of afirst layer (layer A) and a nominal thickness of the first layer, Δt_(B)is a difference in thickness of a second layer (layer B) and a nominalthickness (e.g., a desired thickness, or a thickness provided by auser/designer) of the second layer, Z_(A1) and Z_(A7) are tuningparameters of the third substrate model, Z_(B1) and Z_(B7) are tuningparameters of the fourth substrate model, and R is the radial distanceon the substrate. The substrate model 803 may be evaluated for alllocations on a substrate and the tuning parameters may be determinedbased on the measurement data 801 and the objective function thatincludes a cost function (e.g., a second order polynomial, RMS, MSE,etc.).

In an embodiment, the substrate model 803 that is expressed in polarcoordinates including radius r and Zernike polynomial may be a set ofequations such as the equations 5-8 as follows:

r=√{square root over (x ² +y ²)}  (5)

f1=1   (6)

f4=2r ²−1   (7)

Δthickness=Z1*f1+Z4*f4   (8)

The above substrate model may account for process variations in terms ofZernike polynomials, thus enabling the substrate model to predict moreaccurate stack characteristics (and stack configuration) compared to aconventional methods in which process variation is not taken intoaccount and the stack characteristic (e.g., thickness of a layer) isconsidered as a constant at different locations on the substrate.

In process P802, the tuning parameters may be initialized based on userinputs 802. In an embodiment, the initial values and/or modified values(i.e., the initial values modified during the iterative process) may bedetermined based on, for example, a Monte Carlo based sampling of thevalues of the model parameters. In an embodiment, such Monte Carlosampling may be based on an optimization algorithm that determines asample size based on a search space and an objective function to bereduced, discussed later in the disclosure. It can be appreciated by aperson skilled of ordinary skill in the art that the space of modelparameters may be very large and finding the most appropriate values ishighly challenging and computationally intensive. For example, eachlayer may be associated with a different substrate models based on thestack characteristics (e.g., more than 3) each associated with multiplemodel parameters (e.g., 2 per stack characteristic). Furthermore, thesubstrate may include a large number of layers (e.g., more than 10).Furthermore, each model parameter may take any values in infinite space.Thus, selection of appropriate values of the model parameter and/ordetermination of optimum model parameter values is not trivial.

Furthermore, in process P804, the method involves generating, viasimulation of the substrate model 803 and design for control simulation,a stack map including a plurality of stack configurations, where eachstack configuration is associated with a particular location on thesubstrate. In process P804, the values of tuning parameters determinedin the process P802 may be used in the equations discussed above, alongwith a x-y location values (e.g., a center of the substrate (i.e., 0,0),at an edge of the substrate (e.g., 8 mm, 6 mm), etc.). Thus, based onthe locations, different stack characteristics (and corresponding stackconfigurations) may be determined. Depending on the stack characteristicused, each stack configuration includes specific stack characteristics(e.g., thickness, CD, pitch, etc.) and related values. In an embodiment,the stack configuration at each location (e.g., P1, P2, P3, P4, etc.)includes a thickness of each layer, a critical dimension (CD) of afeature on each layer, an overlay between each layer, etc.

FIG. 10 illustrates an example stack map 1000 including five stackconfigurations 1002, 1004, 1006, 1008, and 1010 at different locationsP1, P2, P3, P4, and P5, respectively, on an example substrate 1001. Eachposition may be measured in terms of Cartesian coordinates or polarcoordinates. Such location information is later used in the process todetermine a fit between the stack model 803 and the measurement data. Inan embodiment, the stack configuration 1002, 1004, 1006, 1008, and 1010may have different stack characteristic due to process variations acrossthe substrate. When the stack configuration is determined based on suchvaried stack characteristics, the model parameters (k1, k2, k3, etc.)determined at the end of the optimization process of the present methodprovide the stack model that is capable of accounting for processvariations across the substrate.

Furthermore, in process P806, the stack characteristics (e.g., the deltavalues Δt_(A), Δt_(B), etc.) may be used as perturbations in asimulation of a patterning process (e.g., D4C) to determine the stackconfiguration or a geometry of a metrology target at differentlocations. In an embodiment, as discussed with respect to FIGS. 7A-7C,in the D4C method individual steps of a lithography process are modeledinto a single process sequence to simulate the physical substrateprocessing. That process sequence drives the creation of the devicegeometry (e.g., stack configuration) as a whole, rather than “building”the device geometry element-by-element. This is different fromconventional approaches that use purely graphical volume elements in athree-dimensional schematic editor to build metrology targets. In anembodiment, the process P806 may further modify the tuning parameters ofthe substrate model 803 to determine the stack configuration. In anembodiment, the processes P804 and P806 may be executed in tandem ortogether to determine the stack map.

The method, in process P808, determining the values of the modelparameters based on a fitting between stack configuration of processP806 and the measurement data such that a cost function (e.g., anobjective function) is reduced. An example cost function is discussed indetail later in the disclosure. In an embodiment, the cost function maybe a mean square error (MSE), a root mean squared error (RMS), or anyother appropriate statistical metric that determines a differencebetween the simulated values of the stack configuration and themeasurement data. In an embodiment, the fitting process may involvemodifying the tuning parameters of the substrate model 803 such that thecost function is reduced (in an embodiment, minimized). The fitting maybe an iterative process, where model parameters of the stack model areiteratively determined so that the stack model is fitted with themeasurement data with high accuracy. In other words, the fitted stackmodel is highly correlated with the measurement data (e.g., as shown inFIGS. 12A-12E).

Further, in process P810, a determination is made whether a stopcriterion is satisfied (e.g., a value of cost function breaches athreshold) or a number of selected iterations is reached. If the stopcriterion is not satisfied, in process P812, an optimization algorithmis executed to determine a next set of value of the model parameter (orintermediate parameter values) from the parameter space to be used inthe next iteration of the method. In an embodiment, the optimizationalgorithm may be based on a gradient-based method (e.g., a gradientdescent method), where a gradient of the cost function is evaluated andthe values of the model parameter that reduces or minimizes the costfunction is selected. In an embodiment, the optimization algorithm maybe a model-based trust region global optimization algorithm.

As an example, a cost function used to optimize stack characteristicsand the stack configuration is expressed in equation 9 below:

$\begin{matrix}{{C{F\left( {z_{1},z_{2},\ldots\mspace{14mu},z_{N}} \right)}} = {\sum\limits_{p = 1}^{P}{w_{p}{f_{p}^{2}\left( {z_{1},z_{2},\ldots\mspace{14mu},z_{N}} \right)}}}} & (9)\end{matrix}$

In equation 9, (z₁, z₂, . . . , z_(N)) there are N design variables(e.g., stack characteristics) or values thereof. In an embodiment,f_(p)(z₁, z₂, . . . , z_(N)) can be a function of the design variables(z₁, z₂, . . . , z_(N)), such as a metric characterizing the degree ofmatching between the result (e.g., thickness, sidewall angle, overlay,alignment, focus) of a particular target design (e.g., a stackconfiguration) as measured using a particular substrate measurementrecipe and the one or more patterns of one or more functional devices,for a set of values of the design variables of (z₁, z₂, . . . , z_(N)).f_(p)(z₁, z₂, . . . , z_(N)) can be a metric (e.g., a key performanceindicate) characterizing the performance (e.g., detectability,printability, sensitivity, stability, etc.) of a particular targetdesign in combination with an associated substrate measurement recipe.In an embodiment, f_(p)(z₁, z₂, . . . , z_(N)) can be a metriccharacterizing the detectability of the particular target design withits associated substrate measurement recipe, namely a measure of theability of the measurement apparatus and process to detect and measurethe particular target design with its associated substrate measurementrecipe. In an embodiment, f_(p)(z₁, z₂, . . . , z_(N)) can be a metriccharacterizing the stability of measurement using the particular targetdesign with its associated substrate measurement recipe, namely how muchthe result of the measurement of the particular target design with itsassociated substrate measurement recipe varies under perturbation. So,in an embodiment, CF(z₁, z₂, . . . , z_(N)) is a combination of af_(p)(z₁, z₂, . . . , z_(N)) characterizing a degree of matching betweenthe result (e.g., a layer thickness, sidewall angle, overlay, alignment,focus) of a particular target design as measured using a particularsubstrate measurement recipe and the behavior of one or more patterns ofone or more functional devices and performance a f_(p)(z₁, z₂, . . . ,z_(N)) characterizing the detectability of the particular target designwith its associated substrate measurement recipe. w_(p) is a weightconstant associated with f_(p)(z₁, z₂, . . . , z_(N)) and of course,could have different values for different f_(p)(z₁, z₂, . . . , z_(N)).Of course, CF(z₁, z₂, . . . , z_(N)) is not limited to the form in Eq.9. CF(z₁, z₂, . . . , z_(N)) can be in any other suitable form.

Thus, in an embodiment, the cost function can include both performanceindicators of device pattern matching and target detectability. In anembodiment, the cost function can be the same, or similar in form to,the following:

$\begin{matrix}{{{Cost}\mspace{14mu}{Function}} = {\sqrt{\left( {W1*{PI}_{{device}\mspace{14mu}{matching}}} \right)^{2} + \left( {W2*{PI}_{detectability}} \right)^{2}} + {{Penalty}\mspace{14mu}{function}\mspace{14mu}\left( {{PI}_{{device}\mspace{14mu}{matching}},{PI}_{detectability}} \right)}}} & (10)\end{matrix}$

In equation 10 above, PI_(device matching) is the performance indicatorfor device pattern matching, PI_(detectability) is the performanceindicator for target detectability, and W1 and W2 are weightingcoefficients. With this format, both device pattern matching and targetdetectability are co-optimized mathematically. If better device patternmatching is desired, then W1 would be larger than W2, for example.

In an embodiment, the cost function for PI_(detectability) comprises√{square root over (TC²+1/SS²)} wherein TC is target coefficient and SSis stack sensitivity.

In one embodiment, the design variables (z₁, z₂, . . . , z_(N)) compriseone or more characteristics/parameters of the target. For example, thedesign variables can include one or more geometric characteristics(e.g., pitch of features of a periodic structure of the target, CD of afeature of a periodic structure of the target (e.g., the widths of theexposed portions and/or unexposed portions), segmentation of individualfeatures of a periodic structure of the pattern, shape of at least partof a periodic structure, length of a periodic structure or of a featureof the periodic structure, etc.) and/or one or more materials properties(e.g., refractive index of a layer of the target, extinction coefficientof a layer of the target, etc.). In an embodiment, the design variablesinclude a plurality of characteristics/parameters of the target. In anembodiment, the design variables can include any adjustable parametersof the substrate measurement recipe. For example, the design variables(z₁, z₂, . . . , z_(N)) may include wavelength, polarization, and/orpupil shape specified in the substrate measurement recipe.

In an embodiment, the stack tuning may also include a process tooptimize a target and/or substrate measurement recipe to make the resultthereof match one or more patterns of a functional device on thesubstrate. Some or all of the parameters of the target and/or substratemeasurement recipe may be adjusted in the optimization. For example, oneor more parameters of the target and/or one or more parameters of themeasurement may be adjusted. The optimization may use a cost functionthat represents a metric characterizing the degree of matching betweenthe result (e.g., overlay, alignment, focus) of using a particulartarget design in combination with a substrate measurement recipe and theone or more patterns of one or more functional devices. As noted above,the result of measuring a target (of a particular design) using asubstrate measurement recipe may be simulated. Thus, in an embodiment,the metric may be a difference between the result and the measurement.The cost function may further represent or be constrained by theperformance (e.g., detectability of the target, printability of thetarget, measurement sensitivity of the target, stability of measurement)of the target in combination with an associated substrate measurementrecipe. Stability is how much the result of using the substratemeasurement recipe to make a measurement with a target varies under aperturbation.

According to an embodiment, the optimization process of stackconfiguration boils down to a process of finding a set of modelparameters of the substrate model 803 that optimizes (e.g., minimizes ormaximizes) the cost function. The cost function can have any suitableform depending on the goal of the optimization. For example, the costfunction can be weighted root mean square (RMS) of deviations of certaincharacteristics of the process and/or system with respect to theintended values (e.g., ideal values) of these characteristics; the costfunction can also be the maximum of these deviations (i.e., worstdeviation). The design variables can be confined to finite ranges and/orbe interdependent due to practicalities of implementations of theprocess and/or system. In the case of a patterning process, theconstraints are often associated with physical properties andcharacteristics of the hardware and/or patterning step, such as tunableranges of hardware and/or patterning device manufacturability designrules.

Physically, the (mis)matching (e.g., overlay shift) is mostly induced byoptical aberrations when printing the device and the target on thesubstrate. How the target is measured (e.g., the target's detection by ameasurement apparatus) will not affect how much the target is shifted.On the other hand, the detectability of the target is determined by theinteraction between upper and lower periodic structures of the target(for an overlay target) or to the interaction between the targetperiodic structure and a sensor (for an alignment target). So, a shiftintroduced by aberrations usually has little or no impact on thedetectability if the target is in the region of good detectability. So,these two effects are somewhat independent of each other, except thatboth will be influenced by the target's characteristics in terms ofgeometry, materials property, etc. So, changing a target characteristiccould have a large impact to one metric but have little impact toanother. So, in an embodiment, having consideration of these properties,an optimizer can find a solution.

Furthermore, in an embodiment, an optimization algorithm, discussedlater in the disclosure, may be applied to select the values of themodel parameter. The optimization algorithm is based on a parabolicfunction that establishes a trend between the measured values and thefitted values. The optimization algorithm has a faster convergence rateand produces accurate results, as well as reduce the computation timecompared to convention optimization algorithms such as gradient decentor trust region global optimization algorithm. The optimizationalgorithm is discussed in detail with respect to FIG. 14. However, thepresent disclosure is not limited by the type of optimization algorithmand any effective optimizer can be used in this method. Furthermore,based on the optimized values of the model parameters, one or more stackcharacteristics (e.g., a layer thickness, SWA, etc.) can be deduced fromthe respective substrate model and the location on the substrate.

When the stop criterion is satisfied, in process P810, the values 810 ofthe model parameters are considered as final or optimized values 810 ofthe model parameter. The optimized values 810 of the model parametersmay be further used (e.g., by a metrology tool) to accurately predict astack configuration (e.g., a metrology target geometry). The improvementoccurs due to the substrate model that has the optimized parametervalues, where the parameter values are determined based on locationspecific stack information (e.g., stack characteristics) andmeasurements, thus capturing the process variation across the entiresubstrate (e.g., via the stack map).

The substrate model 803 with optimized model parameters establishes ahigh level of correlation (e.g., measured in terms of RMS) between themeasurement and modeled data. The correlation may be explained in termsof a key performance parameter (KPI) such as stack sensitivity to achange in model parameter value. FIGS. 12A-12E illustrate examplecorrelations between measured and modeled KPI of different layers atdifferent locations on the substrate.

In FIGS. 12A-12E, the data points are associated with a position and/ora receipe combination of a substrate subjected to the patterningprocess. A high level of correlation is achieved through the tuningprocess of the substrate model 803, which may further help a user toinvestigate a stack configuration and identify the best stack model fora target design. In other words, based on a position on substrate beingmeasured and the recipe corresponding to the position, an accurate stackconfiguration may be generated using the substrate model 803.

In an embodiment, with reference to FIGS. 12A-12E, an optimum (in anembodiment, best) tuning candidate (e.g., a layer of stack of thesubstrate) may be determined using the correlation of each substrateposition and substrate maps of each layer. For example, FIG. 12A-12Eshow the correlations of each target recipe combination for eachsubstrate position (e.g., P1, P2, P3, P4, and P5) in separate graphs.Each position demonstrates a good correlation between the simulated KPIand the measured KPI, thus indicating that the tuning candidate is validfor the tuning of the substrate model 803. For example, in FIG. 12A, thefirst key performance indicator KPI1 corresponding to the measurementdata and the substrate model (e.g., of a first layer) based simulationresults 1201 show a high correlation. Similarly, in FIG. 12B, the secondkey performance indicator KPI2 corresponding to the measurement data andthe substrate model (e.g., of a second layer) based simulation results1202 show a high correlation. In FIG. 12C, the third key performanceindicator KPI3 corresponding to the measurement data and the substratemodel (e.g., of a third layer) based simulation results 1203 show a highcorrelation. In FIG. 12D, the fourth key performance indicator KPI4corresponding to the measurement data and the substrate model (e.g., ofa fourth layer) based simulation results 1204 show a high correlation.In FIG. 12E, the fifth key performance indicator KPIS corresponding tothe measurement data and the substrate model (e.g., of a fifth layer)based simulation results 1205 show a high correlation. Hence, the highcorrelation enables tuning of the one of more of characteristics of thelayers represented by respective substrate models to generate an optimumstack configuration.

The above method provides several advantages. For example, maps of afingerprint or a characteristic (e.g., 1302, 1304, 1306, 1308) ofdifferent layers thickness-deviation from nominal may be generated.FIGS. 13A-13D show the maps of different layers indicating reasonableprocess variations across the different layers of the substrate. In anembodiment, the maps are generated using the substrate model of therespective layers, each model having respective optimized parameters, asdiscussed earlier with respect to the method in FIG. 8. For example, aresist layer 1302 (in FIG. 13A) shows a substantially constant variationin thickness-deviation across the substrate. An etch layer 1304 (in FIG.13B) shows a thickness deviations around the edge of the substrate andat a center of the substrate. For example, the etch layer 1304 has arelatively greater thickness at the edges and a lesser thickness at thecenter of the substrate compared to a nominal thickness of the etchlayer. Similarly, oxide layers 1306 and 1308 (e.g., SOH layer in FIG.13C, and ASI layer in FIG. 13D) show reasonable thickness deviationsacross the substrate. Thus, the substrate model 803 with optimal ortuned model parameters may accurately predict a stack configuration anda target geometry via simulation.

Furthermore, maps of a fingerprint or a characteristic (e.g., in termsof thickness deviation, overlay, alignment, etc.) of a process of thepatterning process or a layer formed by a process on the substrate mayalso be generated. For example, the substrate model 803 may also be usedto generate a fingerprint (e.g., see FIGS. 11A-11D) caused by certainprocesses by simulation of a substrate model specific to a particularlayer generated by the patterning process. For example, a fingerprint1102 represents a thickness of a resist layer across the substrate, afingerprint 1104 represents a thickness of an oxide layer, a fingerprint1106 represent a thickness of the SiN layer, and a fingerprint 1108represent a residual thickness, which is a remaining thickness that maybe determined by removing thickness of different layers from the totalthickness of the stack configuration. Thus, the substrate model canreconstruct fingerprints of each layer which may be further used forcontrolling one or more steps of the patterning process.

Furthermore, the method enables automatic stack tuning or stackconfiguration from any arbitary conditions with capabilities to handle alarge number of stack tuning parameters. Particularly, the locationbased substrate model 803 enables stack tuning of entire substrate thatresults in a high level of matching between the simulation results tothe measurement results.

The stack configuration across the substrate automatically includesprocess variation factors that minimize error and offer accuratemetrology control. In other words, there is no need to identify aposition of interest during a stack reconstruction related to a process.A tool implementing the above method may substantially reducetime-intensive manual work (e.g., manual tuning performed byCS/Field/Customer engineers), which enables engineers to identify theaccurate model for a true stack (i.e., an ideal stack) within a shortertime, hence helps improve the overall product performance. In mostcases, the stack tuning based on the substrate model 803 is far superiorthan the results from tedious trial and error manual work.

The stack-tuning or determination of a stack configuration is a globaloptimization problem. Conventional solutions to such global optimizationproblems have several limitation including (i) local minimums issues,and (ii) computationally expensive global optimum search. Theconventional tools implementing algorithms based on local optimizationsolvers can only find a close by local minimum. Most local optimizationmethods are gradient-based. These algorithms may result in a sub-optimalor untrue stack (i.e., because of use of a local optimum instead of theglobal optimum). Secondly, it is well known that nonlinear globaloptimization is challenging and expensive. The conventional globaloptimization tools are either based on brute force search or line scan.These tools have acceptable performance only when the problem is easyand the number of tuning parameters is small (e.g., less than 5).However, the conventional tool can be extremely time-consuming, unableto handle large number of tuning parameters, and unable to guaranteeglobal optimum due to the nature of the algorithm. These problems withthe conventional tool are further addressed by an optimization methoddiscussed below.

FIG. 10 is a flowchart of an optimization method for determining valuesof model parameters of a model. For example, optimum parameter valuesthat generate improved results when the model is executed. In anembodiment, the model (e.g., the substrate model 803) may be astatistical (or empirical or other mathematical model) representing acharacteristic (e.g., thickness, side wall angle, focus, overlay) of thepatterning process. The method determines values of model parametersbased on an objective function and a fitting between the model andmeasurement data. In an embodiment, the values of the model parametersare determined based on updating a starting point (e.g., a center of asearch region) and a search region (e.g., characterized by a radius) ofthe model parameters based on an objective function such that theobjective function is reduced (in an embodiment, minimized). Theobjective function includes one or more terms including a fit level term(e.g., RMS) such as a measure of error between fitted and measured data,a first penalty term such as a Euclidean distance between the a currentcoordinate and a predicted next coordinate, and a second penalty termthat forces the objective function to have positive values of parametersassociated with the second order terms of the objective function.

In an embodiment, the optimization method may be applied to thesubstrate model 803 such as represented by equations 1-8, discussedabove, to determine values of the model parameters such as k1, k2, k3,k7, Z1, Z2, Z4, etc. Given the number of parameters that must beoptimized (in an embodiment optimized simultaneously), the method of thepresent disclosure can converge to the optimum values of the modelparameters orders of magnitude faster than the conventional optimizationalgorithms such as gradient decent or trust-region. Thus, enablingreal-time execution and improving the productivity of the patterningprocess. Furthermore, in an embodiment, the method may be used offlineor real-time during the patterning process to determine optimum (e.g., alowest value of a cost function) stack configuration. In an embodiment,metrology control and/or patterning process control may be determinedbased on the stack configuration.

The optimization method involves, in process P1401, obtaining (i)initial values includes a starting point and a search region of themodel parameters, and (ii) measurement data corresponding to thecharacteristic (e.g., thickness of a layer, SWA of a layer, metrologyrecipe, etc.) of the patterning process. Furthermore, in an embodiment,the process P1401 may obtain a predicted characteristic (e.g., a stackcharacteristic as discussed earlier) using the initial values of themodel parameter and the measurement data, and (iv) an objectivefunction, discussed later in the disclosure with respect to processP1405. The predicted characteristic refers to a characteristic that themodel is configured to predict. For example, the substrate model 803based on the equations 1-8 are configured to predict a thicknessdifference (i.e., a characteristic) of a layer or multiple layers. In anembodiment, the predicted characteristic is iteratively computed inprocess P1403 of the present method.

In an embodiment, the starting point represents a value corresponding toa center of the search region. The search region is characterized (andmodified) by a radius. The radius defines a limited search region withina potentially infinite space of model parameters. In an embodiment, auser may define constraints on the model parameters or a range of themodel parameters to limit the search space. The center is a point withinthe search region of the model parameters and radius is a distance fromthe center, thus the radius creates an envelope around the center withinwhich sampling may be performed. The center and radius are updatedduring one or more iterations of the optimization process based onpredicted values of the model. For example, the center and/or radius isbased on optimization of a key performance indicator (KPI) or a costfunction determined based on the predicted values of the model. Forexample, the center may be moved to reach a global minima of the costassociated with the model predictions and/or the radius may beincreased, decreased, or maintained at a current value based on afitting level (or a quality of fit) between the measurement data andfitted data predicted by the model.

In an embodiment, the search region is a hyperball representing valuespace related to multiple parameters, where the center and radius of thesearch region are controlled by the model predictions and/or a fittingquality of the model with respect to the measurements. The center and/orradius may be chosen based on the fitting quality and certain criteria,where the criteria are hyper-parameters that a user can define. Thecenter and/or radius may be selected based on different strategies, forexample, based on a high fitting quality (e.g., a fitting quality ratio≥1), a low fitting quality value (e.g., <1) or even slightly negativevalue etc.

The fitting quality (also referred as the fit or the fitting level) maybe characterized by one or more statistical metric such as RMS, MSE, orother appropriate data fitting metrics. The fitted data is datapredicted by the model (e.g., the substrate model 803) based on thevalues of the model parameter within the search region. The center andthe radius may be modified based on a quality of fit between the fitteddata and the measurement data, and penalty terms embedded in anobjective function, as discussed in detail below. Such updating of thecenter and radius moves the search region from a random starting pointin a potentially infinite space to a global minima or other optimumvalues of the model parameter for which the objective function relatedto the model is reduced (in an embodiment, minimized).

Furthermore, in the process P1401, the starting point and the searchregion is used to draw a first sample or a first set of sample points(e.g., 10 points, 20 points, 30 points, etc.) from the parameter space.For example, the first sample may be drawn by a sampling method (e.g., aMonte Carlo based method) based on a certain probability distribution(e.g., a uniform probability distribution. a normal probabilitydistribution, or other probability distributions) within a search region(e.g., 1722 in FIG. 17A). In an embodiment, the search region may bemodified in a subsequent iteration and a second sample may be drawn. Thesearch region may be modified by modifying, for example, the center andthe radius of the search region, further illustrated in FIGS. 17A-17C.Thus, the sampling is a self-adaptive process that updates the samplesin an iterative manner. Such adaptive sampling reduces a computationtime (or simulation time) of a patterning process (e.g., the D4Csimulation), thus improving the productivity of the patterning process.In an embodiment, the patterning process (e.g., the D4C simulation) maybe a real-time simulation product which runs very slowly in case oflarge number of simulations. Hence, the optimization method, accordingto the present disclosure, substantially reduces the amount ofsimulations and makes efficient and reasonable sampling possible formodel fitting (e.g., fitting the substrate model 803 to the measurementdata).

In an embodiment, a sampling scheme is characterized by a number ofsamples (or sampling density) drawn from the search region as a functionof number of parameters and the radius (which can becontrolled/updated). Accordingly, the number of samples may change witha change in the radius during the iterative process. In an embodiment,the number of samples may be controlled to improve the efficiency andaccuracy of results of the method. In an embodiment, the number ofsamples (S) may be determined based on following equation 11:

$\begin{matrix}{S = {{SR}*{number}\mspace{14mu}{of}\mspace{14mu}{parameters}*\left( {{A*e^{- \frac{{radiu}s^{2}}{2*B^{2}}}} + C} \right)}} & (11)\end{matrix}$

In the above equation, SR is a sample ratio, which is a hyperparameter,and A, B, and C are control parameters of the number of samples (S).radius is the search radius of the parameter space. In an embodiment,the number of samples can be updated by tuning the sample ratio (SR). Inan embodiment, the sample ratio (SR) provides an external control to theuser to modify the number of samples. Also, the number of samples may becontrolled by changing the values of parameters A, B, and/or C. However,changing parameters A, B, and/or C may be based on understanding of thebehavior of the exponential functions. In an embodiment, such controlparameters A, B, and/or C may be changed less frequently compared to thesample ratio.

In a non-limiting example of the sampling scheme as a function of theradius is illustrated in a graph 1502 of FIG. 15. The graph 1502 showsthat the number of samples increase as the radius increases.Accordingly, during an iteration of the method, the number of samplesmay increase if the radius of the search region is increased, or thenumber of samples may decrease if the radius of the search region isdecreased. The increasing or decreasing of the radius depends on thepredicted values of the model and/or the quality of fit, as discussedlater in the disclosure (e.g., in FIGS. 17A-17C).

Hence, using the equation 11 above, the sampling can be controlled basedon the number of model parameters and the radius. Furthermore, thesampling may be controlled by varying the control parameters A, B,and/or C, or the hyper-parameter sample ratio. In subsequent iterations,the radius and/or the center may change based on the objective function.For example, the center may be moved to a point within the search region(or slightly outside the search region) where the objective function hasa relatively low value, further illustrated with respect to FIGS.17A-17C. Thus, the sample scheme is adapting with change in the radiusand the center. In an embodiment, a current sample (e.g., a firstsample) may be included in a sample pool and reused during iterations,for example, to determine the center and/or radius. This furtherimproves the efficiency of the optimization method or the optimizationalgorithm.

Furthermore, the values of the model parameters (e.g., the first sample)may be used to evaluate the model (e.g., thickness of a layer, SWA,etc.). Further, the process P1403 involves determining a fitting betweenthe model and the measurement data. For example, the model fitting maybe determined based on the first sample and a cost model. In anembodiment, a cost model may be a defined as, for example, a secondorder polynomial model. The second order polynomial model may be aparabolic model that captures an overall trend of the data. In addition,the second order polynomial model is noise-resistive and is not affectedby local minimums of the search space or the entire space in general.Thus, compared to the traditional gradient based methods, the method isnoise-resistive and not affected by local minimums. In other words, ifthe solution space includes many small local minimums/fluctuations, theparabolic fitting can ignore such local minimums/fluctuations.

An example fitting of measurement data and model is illustrated in FIGS.16A-16D. In the non-limiting example, FIGS. 16A-16D illustrates afitting of the model with measurements of SWA of an etch layer collectedfor measurement targets 1, 2, and 3 located at different locations on asubstrate. In FIG. 16A, the fitting is based on a 30 sample points. Acost or error (e.g., a distance or RMS value) between the fitted model(i.e., a curve 1602) and the measurement data 1601 varies as the SWAincreases or decreases and the cost appears to be lowest for side wallangle around or at zero. The cost or error increases as the SWA becomesgreater than zero or lower than zero. It can be seen that the parabolicmodel fitting (i.e., the curve 1602) captures the overall trend of themeasurement data 1601. As the SWA increases from a negative value tozero, the cost or error decreases and forms a valley around the zerovalue (e.g., between −1 and 5 units). Further, as the SWA increasesbeyond 5, the cost or error starts increasing.

In an embodiment, FIG. 16B illustrates a similar fitting between curve1611 and the measurement data 1610. The curve 1611 also captures thetrend with fewer data points (e.g., 20 data points) than used in FIG.16A. Similarly, FIG. 16C illustrates a fitting of the model (i.e., acurve 1631) with measurements data 1630 of SWA of an etch layercollected for measurement targets 4, 5, and 6 located at differentlocations on the substrate. It can be seen that the parabolic modelfitting (i.e., the curve 1631) captures the overall trend of the data1630. A valley or low cost area is formed around the SWA value of 10. Inan embodiment, FIG. 16D illustrates a similar fitting between a curve1641 and the measurement data 1640. The curve 1641 also captures thetrend with fewer data points (e.g., 20 data points) than used in FIG.16D.

The parabolic model fitting captures the overall trend of the data. Theparabolic model helps to determine a most likely location of a costvalley (e.g., where an error between the fit and measured data isreduced or minimum). Being able to locate the cost valley enables modelfitting using fewer sample. For example, even if sampling around onlypart of the valley, the fitting can still direct towards the overallvalley orientation. Such a direction is an approximate direction but asthe model parameter value progressively leads to a global minimum of theobjective function, the solution space of the model parameter becomesmore prominent. Thus, some of the benefits of using a parabolic model,according to an embodiment, includes faster convergence compared togradient descent method, where finding the next point and/or searchregion is based on a certain learning rate (i.e., a small increment). Onthe other hand, the parabolic model enables a jump with a bigger step(compared to the learning rate) from one search area to another area.Furthermore, the parabolic model provides a holistic view (instead oflocal information as in a gradient descent method) and can be noiseresistive, as mentioned earlier.

In an embodiment, the model can be represented by equation 12 asfollows:

Y=(a ₁ X ₁ +b ₁)²+(a ₂ X ₂ +b ₂)²+(a ₃ X ₃ +b ₃)²+ . . . +(a _(n) X _(n)+b _(n))²   (12)

In the above equation 12, X₁, X₂, X₃, . . . , X_(n) represent acoordinate vector in n-parameter space, Y is a characteristic to bepredicted, and a₁ through a_(n) and b₁ through b_(n) are the modelparameters to be determined. In the above model, the prediction ofcharacteristic Y is based on a global minimum value i.e., Y=0, whichindicates a coordinate X_(n) is a function of model parameter, e.g.,

$X_{n} = {- {\frac{a_{n}}{b_{n}}.}}$

In an embodiment, the working principle of fitting the above model (Y)can be explained as follows. Assume (i) there exists a function thatdescribes a contribution to an error in model prediction ascost_(i)(Y)=f(X_(i)), and (ii) all model parameters are independentcontributing factors (e.g., X₁, X₂, X₃, . . . , X_(n)) to a disturbanceto a nominal stack (e.g., a true stack provided by a designer) and thatthese factors induce error in a KPI (e.g., stack sensitivity), then anoverall cost of each point in the multi-dimensional parameter hyperspace(e.g., having more than 4 dimensions or model parameters) is a linearcombination of contribution of each individual model parameter that canbe represented as equation 13 below:

$\begin{matrix}{Y = {{{a_{1}{f\left( X_{1} \right)}} + {a_{2}{f\left( X_{2} \right)}} + {a_{3}{f\left( X_{3} \right)}} + \ldots + {a_{n}{f\left( X_{n} \right)}}} = {\sum\limits_{i = 1}^{n}{a_{i}{f\left( X_{i} \right)}}}}} & (13)\end{matrix}$

In an embodiment, f(X_(i))=(a_(i)X_(i)+b_(i))² is a parabolicrelationship used to approximate the error behavior of a disturbance ora perturbation. For example, FIGS. 16A-16D illustrates an error behaviorwith respect to a disturbance (e.g., in the SWA). FIG. 16A indicatesthat a large perturbation causes a large error to be added to the cost.In an embodiment, the error is treated symmetrically when theperturbation goes to either positive or negative direction (e.g.,positive or negative values of SWA). Thus, the model is composed of anetwork of 2nd order polymial functions (e.g., a parabolic function)Y=Σ_(i=1) ^(n)Y_(i)=Σ_(i=1) ^(n)(a_(i)X_(i)+b_(i))², where each of termsrepresents a cost associated with a characteristic of the substrate(e.g., SWA, thickness) in terms of the multi-dimensional model parameterspace. In an embodiment, the cost Y_(i) symmetrically distributes aroundX_(i) position, which is computed as

${- \frac{b_{i}}{a_{i}}},$

where Y or Y_(i) has the lowest value (e.g., Y=0).

In an embodiment, the model (Y) is based on two assumptions: (i) theerror each perturbation or disturbance contributes has a parabolicrelation with its perturbation amplitude, and (ii) contributions ofmodel parameters are independent of each other. In several cases the twoassumptions may be violated, however, the model still enables goodpredictions because the model captures the overall trend of a hyperspaceof the model parameters without overfitting or overcompensating forcomplicated curvature of the hyperspace. Of course the parabolic modelis not the best approximation function, however the parabolic model is asimple to implement and enables improved computation speed and resultingvalues of the parameters provide high correlationship between themeasurement data and the model.

In an embodiment, the above model may be used in conjunction with anobjective function (or a cost function) in a trust region-likealgorithm. The objective function used within the trust region algorithmincludes penalty term(s) according to the present model. In anembodiment, the objective function comprises more than one termincluding a fitting term (e.g., RMS), a first penalty term (e.g., aneculidean distance), and/or a second penalty term (e.g., penaltypositive). The objective function according used for optimization methodmay be represented by by following equation 14:

Objective=fit level+λ1*penalty_distance+λ2*penalty_positive   (14)

In the above equation 14, the objective is a value to be reduced (in anembodiment, minimized), a fit level is a term (e.g., RMS, MSE, etc.)that determines a level of fit between the model and the measured data,λ1 and λ2 are parameters that can be optimized to improve the penaltyperformance, pentalty_distance is a euclidean distance between the lastcoordinate (the starting point for a current iteration) and thepredicted next coordinate (e.g., center calculated in a currentiteration), and the penalty_positive term is used to force thecoefficients of 2nd order terms to be positive. In an embodiment, thepenalty_distance is minimized in order to limit the step size betweenthe starting point and the next point. In several cases once thesolution space has flat or complicated topography, the predicted optimumtends to make large jumps or go to infinity. The penalty_distance termthus guides the model to choose a nearest feasible point (e.g., withinthe search region) with respect to the starting point instead of arelatively faraway low-cost point (e.g., outside the search region).

As mentioned earlier, the penalty_positive term is used to force thecoefficients of 2nd order terms to be positive. For example, thecoefficients of 2nd order terms may be terms of the model (Y) (e.g.,equation 12) rewritten in an expanded form (equation 15) as follows.Then, the coefficients of 2nd order terms are A1, A₂, A₃, . . . , A_(n)should be positive so that a valley may be formed (e.g., see curves1601, 1611, 1631, and 1641), otherwise the curve will be inverted and novalley may be formed.

Y=A ₁ X ₁ ² +B ₁ X ₁ +C ₁ +A ₂ X ₂ ² +B ₂ X ₂ +C ₂ + . . . +A _(n) X_(n) ² +B _(n) X _(n) +C _(n)   (15)

Furthermore, the above equation 15 may be represented in a matrix form(eq. 16) to build a linear regression model as:

Y=βX   (16)

In the above equation 16, Y is the cost vector, X is generally referredas a design matrix (e.g., representing substrate characteristics such asthickness and SWA) and β is the coefficient vector. An example matrixform is shown below:

$\begin{bmatrix}Y_{0} \\Y_{1} \\Y_{2} \\\vdots \\Y_{m}\end{bmatrix} = {\begin{bmatrix}X_{1,0}^{2} & X_{1,0} & X_{2,0}^{2} & X_{2,0} & \ldots & X_{n,0}^{2} & X_{n,0} & 1 \\X_{1,1}^{2} & X_{1,1} & X_{2,1}^{2} & X_{2,1} & \ldots & X_{n,1}^{2} & X_{n,1} & 1 \\X_{1,2}^{2} & X_{1,2} & X_{2,2}^{2} & X_{2,2} & \ldots & X_{n,2}^{2} & X_{n,2} & 1 \\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\X_{1,m}^{2} & X_{1,m} & X_{2,m}^{2} & X_{2,m} & \ldots & X_{n,m}^{2} & X_{n,m} & 1\end{bmatrix}\begin{bmatrix}A_{1} \\B_{1} \\A_{2} \\B_{2} \\\vdots \\A_{n} \\B_{n} \\C\end{bmatrix}}$

In the above matrix, m is the number of sample points, n is the numberof parameters. A, B correspond to the coefficient of 2^(nd) order and1^(st) order terms, respectively, and C is the intercept. When solvingthis linear matrix equation, it should be ensured that the fittingcoefficient A₁ through A_(n) are all positive numbers in order for themodel to have a global minimum. Negative values of A₁ through A_(n) mayinvert the parabolic curve (e.g., the curves 1601, 1611), and no valleymay be observed. Such constraints may be implemented as inequalityconstraints to form a constrained optimization problem.

In an embodiment, the penalty_positive term of equation 14 is defined asan exponential decay function as follows:

penalty_positive=Σ_(i) ^(n)α₁ *e ^(−α) ² ^(A) ^(i)   (17)

In the above equation 17, α₁ and α₂ are relatively large numbers (e.g.,of the order of 1E10) that can be tuned as an inner variable (e.g.,within a program code) of the optimization algorithm. If any coefficientA₁ through A_(n) goes to the negative value, the penalty_positive willsmoothly shoot to a very large number, effectively forming a barrier toprevent A_(n) from going to the negative value. In other words, a highvalue of the penalty_positive will result in a higher value of theobjective function, thus indicating a non-optimal solution.

In process P1405, the method involves determining a fit quality based onthe objective function. In an embodiment, the fit quality may be definedas a ratio of a modeled cost and a true cost. In an embodiment, the fitquality is modified based on the true cost and the modeled cost torepresent a percentage value. The true cost refers to cost determinedbased on a difference between a reference stack (or true stack/idealstack) provided by a designer and the measurement data. The modeled costrefers to a value of the objective function.

In an embodiment, suppose a model is f(x), the fittingquality=[f(prediction point)−f(last point)]/[actual_value(predictionpoint)−actual_value(last point)].

The fitting quality describes how much the model follows a real shape ofthe solution space. Ideally, the model should follow the shape closely,so that the ratio is approximately 1 or even larger. The smaller theratio the worse the fitting is. If the ratio is negative, it means themodel trend is opposite to the true shape then the fitting is very bad.

Based on the fit quality, further processes involve updating thestarting point and/or the search region. In an embodiment, the radiusand/or the center may be updated. For example, if the fit quality isgood, the center may be updated and the radius may be enlarged. If thefit quality is acceptable, then only center may be updated and theradius may be maintained at a current value. If the fit quality is bad,then the radius may be reduced and the center may be maintained at thecurrent value. In embodiment, a goodness or acceptability of the fitquality may be based on breaching of certain threshold values. Forexample, if the fit quality is greater than, for example 70%, then thefit quality is good. If the fit quality is between 40%-70%, then the fitquality is acceptable. If the fit quality is less than 40%, then the fitquality is bad.

In process P1408, a determination may be made whether the fit qualitybreaches a first threshold. In an embodiment, the first threshold may be70%. Thus, if the fit quality is greater than or equal to 70%, then thefit quality is good and a process P1418 is performed. A good fit qualityindicates that the current search region provides more than sufficientsample points around the current center. As such, the current center maybe moved, as well as the radius may be enlarged.

The process P1418 involves updating the starting point and the searchregion by selecting a new starting point (i.e., a center) and increasingthe search region. In an embodiment, the new center may be a pointhaving a relatively low value of the objective function within thesearch region. In an embodiment, the new center may be a value justoutside the boundary of the search region, in a direction where thevalue of the objective function gradually decreases. Furthermore, in anembodiment, the search region may be enlarged by an enlargement factor.In an embodiment, the enlargement factor may be expressed in terms of apercentage. For example, the search region may be enlarged by increasingthe radius, for example, increasing by 25% of the current radius value.The present disclosure is not limited to a particular enlargement factorand any appropriate enlargement value or an enlargement function may bedefined to gradually increase the search region in a current orsubsequent iterations.

In process P1409, a determination may be made whether the fit qualitybreaches a second threshold. In an embodiment, the second threshold maybe a range between 40%-70%. Thus, if the fit quality is greater than orequal to 40% and less than 70%, then the fit quality is acceptable and aprocess P1419 is performed. An acceptable fit quality indicates that thecurrent search region provides sufficient sample points around thecurrent center. As such, the current center may be moved withoutchanging the radius.

The process P1419 involves updating the starting point by selecting anew starting point (i.e., a center). In an embodiment, the search regionmay not be updated. For example, a current value of the radius may beused for subsequent iteration. In an embodiment, the new center may be apoint having a relatively low value of the objective function within thesearch region. In an embodiment, the new center may be a value justoutside the boundary of the search region, in a direction where thevalue of the objective function gradually decreases.

In process P1410, a determination may be made whether the fit qualitybreaches a third threshold. In an embodiment, the first threshold may be40%. Thus, if the fit quality is less than 40%, then the fit quality isbad and a process P1420 is performed. A bad fit quality indicates thatthe current search region does not sufficient sample points and/or thecurrent center is far from an optimum value of the objective function.As such, the search region may be reduced and the current center may bemaintained.

The process P1420 involves updating the starting point and the searchregion by decreasing the size of the search region. In an embodiment,the search region may be decreased by a reduction factor. In anembodiment, the reduction factor may be in percentage. For example, thesearch region may be decreased by decreasing the radius, for example, by25% of the current radius value. The present disclosure is not limitedto a particular reduction factor and any appropriate reduction value ora reduction function may be defined to gradually decrease the searchregion in a current or subsequent iterations.

In process P1422, a determination is made whether a stopping criterionis met. The stopping criterion may be a threshold value of number ofiterations, or a cost/objective function related value. When thestopping criterion is not met, the flow leads to the process P1401 tostart a next iteration. In the next iteration, the center and radiusvalues determined in the process P1418, P1419, or P1420 are used. Afterseveral iteration, the solution may converge, i.e., no furtherimprovement in the cost or objective function may be observed. When thestopping criterion is met, the model parameter values obtained aretermed as optimized model parameter values, which can be further used todetermine a optimum stack configuration and/or a stack characteristics.

FIGS. 17A-17C illustrate examples of how the radius and center may beupdated. For illustration purposes, a search region 1712 having a center1711 is plotted on an objective function map 1700. The objectivefunction map 1700 graphically depicts regions or a range of values ofthe objective function, for example, a light grey region indicates lowervalues than a dark grey region. In an embodiment, contour lines may alsobe included to indicate the similar values of the objective function.For example, innermost contour lines 1702 and 1703 indicate lower values(in an embodiment, lowest) compared to outer contour lines 1713, 1714,and 1716. In an embodiment, the values of the objective functiongradually increase from inside (e.g., 1715) to outside (e.g., towards1713), each contour line indicating a particular value of the objectivefunction.

In FIG. 17A, once a plot of the objective function is generated (e.g.,via simulation), a first center 1711 (or a starting point) and a firstsearch region 1712 is plotted on the objective map 1700. Within thefirst region 1712, a cost may be estimated using the second orderparabolic equation as discussed above. Based on the cost, a second point1721 may be selected that has a lowest cost value in the search region1712. It may be evaluated that the model parameters corresponding to thesecond point 1721 result in a good fit (e.g., as discussed in theprocess P1408 earlier). Then, the starting point 1711 may be updated asthe second point 1721, as well as the first search region 1712 may beexpanded to a second search region 1722, as mentioned in process P1418earlier. Furthermore, the cost function may be evaluated within thesecond search region 1722 to determine a third center 1731 having alowest cost value within the second search region 1722. In addition, athird search region greater than the second search region 1722 may bedetermined.

In FIG. 17B, illustrates an example, where a bad fit may be obtained andthe search radius may be reduced, as mentioned in processes P1410 and1720 earlier. For example, the third center 1731 and a third searchregion 1732 provide parameter values that result in a bad fit. In thiscase, the third center 1731 may be maintained (i.e., a fourth center issame as the third center 1731) and a fourth search region 1742, which issmaller than the third region 1732 may be determined. Further, the costfunction may be evaluated within the fourth search region 1742 todetermine a fifth center 1751 having a lowest cost value within thefourth search region 1742. It can be seen that the fourth search region1742 provides a next center 1751 close to the low value 1703 (in anembodiment, lowest or minimum value) of the objective function. Thefourth region 1742 and the center 1751 may result in an acceptable fit.

FIG. 17C illustrates an example where the solution converges. Forexample, according to the process P1009 and P1019, the fifth center 1751may be maintained and a fifth search region 1752 may be defined. Themodel parameters within the search region 1752 may result in a good fit.Then, the process of P1008 and P1018 (as illustrated in FIG. 17A) may berepeated, which may result in a sixth center 1761 and a sixth searchregion 1762 that provides a global optimum values of the modelparameter. Any further iteration may not reduce the value of theobjective function, at which point the solution is said to be convergedand the corresponding values of the model parameters are considered asthe optimum values.

In an embodiment, the optimization method may be supplemented withadditional complementary solutions to improve the accuracy andefficiency. For example, a sample pool may be created to reuse certainsamples (e.g., a second sample) to improve efficiency. A brute forcesearch (e.g., as a follow up after the convergence according to thepresent method) to determine an improved parameter values mayalternatively be used. In such brute force based implementation, theresults of the present optimization method may serve as an initialstarting point (instead of a random starting point) that eventuallyleads to faster execution of the traditional brute force based method.The brute force method may also be based on a Monte Carlo samplingmethod.

FIG. 18 is a block diagram that illustrates a computer system 100 whichcan assist in implementing the methods, flows or the apparatus disclosedherein. Computer system 100 includes a bus 102 or other communicationmechanism for communicating information, and a processor 104 (ormultiple processors 104 and 105) coupled with bus 102 for processinginformation. Computer system 100 also includes a main memory 106, suchas a random access memory (RAM) or other dynamic storage device, coupledto bus 102 for storing information and instructions to be executed byprocessor 104. Main memory 106 also may be used for storing temporaryvariables or other intermediate information during execution ofinstructions to be executed by processor 104. Computer system 100further includes a read only memory (ROM) 108 or other static storagedevice coupled to bus 102 for storing static information andinstructions for processor 104. A storage device 110, such as a magneticdisk or optical disk, is provided and coupled to bus 102 for storinginformation and instructions.

Computer system 100 may be coupled via bus 102 to a display 112, such asa cathode ray tube (CRT) or flat panel or touch panel display fordisplaying information to a computer user. An input device 114,including alphanumeric and other keys, is coupled to bus 102 forcommunicating information and command selections to processor 104.Another type of user input device is cursor control 116, such as amouse, a trackball, or cursor direction keys for communicating directioninformation and command selections to processor 104 and for controllingcursor movement on display 112. This input device typically has twodegrees of freedom in two axes, a first axis (e.g., x) and a second axis(e.g., y), that allows the device to specify positions in a plane. Atouch panel (screen) display may also be used as an input device.

According to one embodiment, portions of one or more methods describedherein may be performed by computer system 100 in response to processor104 executing one or more sequences of one or more instructionscontained in main memory 106. Such instructions may be read into mainmemory 106 from another computer-readable medium, such as storage device110. Execution of the sequences of instructions contained in main memory106 causes processor 104 to perform the process steps described herein.One or more processors in a multi-processing arrangement may also beemployed to execute the sequences of instructions contained in mainmemory 106. In an alternative embodiment, hard-wired circuitry may beused in place of or in combination with software instructions. Thus, thedescription herein is not limited to any specific combination ofhardware circuitry and software.

The term “computer-readable medium” as used herein refers to any mediumthat participates in providing instructions to processor 104 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media, volatile media, and transmission media.Non-volatile media include, for example, optical or magnetic disks, suchas storage device 110. Volatile media include dynamic memory, such asmain memory 106. Transmission media include coaxial cables, copper wireand fiber optics, including the wires that comprise bus 102.Transmission media can also take the form of acoustic or light waves,such as those generated during radio frequency (RF) and infrared (IR)data communications. Common forms of computer-readable media include,for example, a floppy disk, a flexible disk, hard disk, magnetic tape,any other magnetic medium, a CD-ROM, DVD, any other optical medium,punch cards, paper tape, any other physical medium with patterns ofholes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip orcartridge, a carrier wave as described hereinafter, or any other mediumfrom which a computer can read.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 104 forexecution. For example, the instructions may initially be borne on amagnetic disk of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 100 canreceive the data on the telephone line and use an infrared transmitterto convert the data to an infrared signal. An infrared detector coupledto bus 102 can receive the data carried in the infrared signal and placethe data on bus 102. Bus 102 carries the data to main memory 106, fromwhich processor 104 retrieves and executes the instructions. Theinstructions received by main memory 106 may optionally be stored onstorage device 110 either before or after execution by processor 104.

Computer system 100 may also include a communication interface 118coupled to bus 102. Communication interface 118 provides a two-way datacommunication coupling to a network link 120 that is connected to alocal network 122. For example, communication interface 118 may be anintegrated services digital network (ISDN) card or a modem to provide adata communication connection to a corresponding type of telephone line.As another example, communication interface 118 may be a local areanetwork (LAN) card to provide a data communication connection to acompatible LAN. Wireless links may also be implemented. In any suchimplementation, communication interface 118 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

Network link 120 typically provides data communication through one ormore networks to other data devices. For example, network link 120 mayprovide a connection through local network 122 to a host computer 124 orto data equipment operated by an Internet Service Provider (ISP) 126.ISP 126 in turn provides data communication services through theworldwide packet data communication network, now commonly referred to asthe “Internet” 128. Local network 122 and Internet 128 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 120 and through communication interface 118, which carrythe digital data to and from computer system 100, are exemplary forms ofcarrier waves transporting the information.

Computer system 100 can send messages and receive data, includingprogram code, through the network(s), network link 120, andcommunication interface 118. In the Internet example, a server 130 mighttransmit a requested code for an application program through Internet128, ISP 126, local network 122 and communication interface 118. Onesuch downloaded application may provide all or part of a methoddescribed herein, for example. The received code may be executed byprocessor 104 as it is received, and/or stored in storage device 110, orother non-volatile storage for later execution. In this manner, computersystem 100 may obtain application code in the form of a carrier wave.

FIG. 19 is essentially identical to the system of FIG. 1 and shown hereagain with certain portions enlarged and others omitted for convenience.FIGS. 1 and 19 depict an exemplary lithographic projection apparatusused in conjunction with the techniques described herein. The apparatus(e.g., see FIG. 19) comprises:

-   -   an illumination system IL, to condition a beam B of radiation.        In this particular case, the illumination system also comprises        a radiation source SO;    -   a first object table (e.g., patterning device table) MT provided        with a patterning device holder to hold a patterning device MA        (e.g., a reticle), and connected to a first positioner to        accurately position the patterning device with respect to item        PS;    -   a second object table (substrate table) WT provided with a        substrate holder to hold a substrate W (e.g., a resist-coated        silicon wafer), and connected to a second positioner to        accurately position the substrate with respect to item PS;    -   a projection system (“lens”) PS (e.g., a refractive, catoptric        or catadioptric optical system) to image an irradiated portion        of the patterning device MA onto a target portion C (e.g.,        comprising one or more dies) of the substrate W.

As depicted herein, the apparatus is of a transmissive type (i.e., has atransmissive patterning device). However, in general, it may also be ofa reflective type, for example (with a reflective patterning device).The apparatus may employ a different kind of patterning device toclassic mask; examples include a programmable mirror array or LCDmatrix.

The source SO (e.g., a mercury lamp or excimer laser, LPP (laserproduced plasma) EUV source) produces a beam of radiation. This beam isfed into an illumination system (illuminator) IL, either directly orafter having traversed conditioning means, such as a beam expander Ex,for example. The illuminator IL may comprise adjusting means AD forsetting the outer and/or inner radial extent (commonly referred to asσ-outer and σ-inner, respectively) of the intensity distribution in thebeam. In addition, it will generally comprise various other components,such as an integrator IN and a condenser CO. In this way, the beam Bimpinging on the patterning device MA has a desired uniformity andintensity distribution in its cross-section.

It should be noted with regard to FIG. 19 that the source SO may bewithin the housing of the lithographic projection apparatus (as is oftenthe case when the source SO is a mercury lamp, for example), but that itmay also be remote from the lithographic projection apparatus, theradiation beam that it produces being led into the apparatus (e.g., withthe aid of suitable directing mirrors); this latter scenario is oftenthe case when the source SO is an excimer laser (e.g., based on KrF, ArFor F₂ lasing).

The beam PB subsequently intercepts the patterning device MA, which isheld on a patterning device table MT. Having traversed the patterningdevice MA, the beam B passes through the lens PL, which focuses the beamB onto a target portion C of the substrate W. With the aid of the secondpositioning means (and interferometric measuring means IF), thesubstrate table WT can be moved accurately, e.g. so as to positiondifferent target portions C in the path of the beam PB. Similarly, thefirst positioning means can be used to accurately position thepatterning device MA with respect to the path of the beam B, e.g., aftermechanical retrieval of the patterning device MA from a patterningdevice library, or during a scan. In general, movement of the objecttables MT, WT will be realized with the aid of a long-stroke module(coarse positioning) and a short-stroke module (fine positioning), whichare not explicitly depicted in FIG. 19. However, in the case of astepper (as opposed to a step-and-scan tool) the patterning device tableMT may just be connected to a short stroke actuator, or may be fixed.

The depicted tool can be used in two different modes:

-   -   In step mode, the patterning device table MT is kept essentially        stationary, and an entire patterning device image is projected        in one go (i.e., a single “flash”) onto a target portion C. The        substrate table WT is then shifted in the x and/or y directions        so that a different target portion C can be irradiated by the        beam PB;    -   In scan mode, essentially the same scenario applies, except that        a given target portion C is not exposed in a single “flash”.        Instead, the patterning device table MT is movable in a given        direction (the so-called “scan direction”, e.g., the y        direction) with a speed v, so that the projection beam B is        caused to scan over a patterning device image; concurrently, the        substrate table WT is simultaneously moved in the same or        opposite direction at a speed V=Mv, in which M is the        magnification of the lens PL (typically, M=1/4 or 1/5). In this        manner, a relatively large target portion C can be exposed,        without having to compromise on resolution.

FIG. 20 schematically depicts another exemplary lithographic projectionapparatus 1000 in conjunction with the techniques described herein canbe utilized.

The lithographic projection apparatus 1000 comprises:

-   -   a source collector module SO    -   an illumination system (illuminator) IL configured to condition        a radiation beam B (e.g. EUV radiation).    -   a support structure (e.g. a patterning device table) MT        constructed to support a patterning device (e.g. a mask or a        reticle) MA and connected to a first positioner PM configured to        accurately position the patterning device;    -   a substrate table (e.g. a wafer table) WT constructed to hold a        substrate (e.g. a resist coated wafer) W and connected to a        second positioner PW configured to accurately position the        substrate; and    -   a projection system (e.g. a reflective projection system) PS        configured to project a pattern imparted to the radiation beam B        by patterning device MA onto a target portion C (e.g. comprising        one or more dies) of the substrate W.

As here depicted, the apparatus 1000 is of a reflective type (e.g.employing a reflective patterning device). It is to be noted thatbecause most materials are absorptive within the EUV wavelength range,the patterning device may have multilayer reflectors comprising, forexample, a multi-stack of Molybdenum and Silicon. In one example, themulti-stack reflector has a 40 layer pairs of Molybdenum and Siliconwhere the thickness of each layer is a quarter wavelength. Even smallerwavelengths may be produced with X-ray lithography. Since most materialis absorptive at EUV and x-ray wavelengths, a thin piece of patternedabsorbing material on the patterning device topography (e.g., a TaNabsorber on top of the multi-layer reflector) defines where featureswould print (positive resist) or not print (negative resist).

Referring to FIG. 20, the illuminator IL receives an extreme ultraviolet radiation beam from the source collector module SO. Methods toproduce EUV radiation include, but are not necessarily limited to,converting a material into a plasma state that has at least one element,e.g., xenon, lithium or tin, with one or more emission lines in the EUVrange. In one such method, often termed laser produced plasma (“LPP”)the plasma can be produced by irradiating a fuel, such as a droplet,stream or cluster of material having the line-emitting element, with alaser beam. The source collector module SO may be part of an EUVradiation system including a laser, not shown in FIG. 20, for providingthe laser beam exciting the fuel. The resulting plasma emits outputradiation, e.g., EUV radiation, which is collected using a radiationcollector, disposed in the source collector module. The laser and thesource collector module may be separate entities, for example when a CO2laser is used to provide the laser beam for fuel excitation.

In such cases, the laser is not considered to form part of thelithographic apparatus and the radiation beam is passed from the laserto the source collector module with the aid of a beam delivery systemcomprising, for example, suitable directing mirrors and/or a beamexpander. In other cases the source may be an integral part of thesource collector module, for example when the source is a dischargeproduced plasma EUV generator, often termed as a DPP source.

The illuminator IL may comprise an adjuster for adjusting the angularintensity distribution of the radiation beam. Generally, at least theouter and/or inner radial extent (commonly referred to as σ-outer andσ-inner, respectively) of the intensity distribution in a pupil plane ofthe illuminator can be adjusted. In addition, the illuminator IL maycomprise various other components, such as facetted field and pupilmirror devices. The illuminator may be used to condition the radiationbeam, to have a desired uniformity and intensity distribution in itscross section.

The radiation beam B is incident on the patterning device (e.g., mask)MA, which is held on the support structure (e.g., patterning devicetable) MT, and is patterned by the patterning device. After beingreflected from the patterning device (e.g. mask) MA, the radiation beamB passes through the projection system PS, which focuses the beam onto atarget portion C of the substrate W. With the aid of the secondpositioner PW and position sensor PS2 (e.g. an interferometric device,linear encoder or capacitive sensor), the substrate table WT can bemoved accurately, e.g. so as to position different target portions C inthe path of the radiation beam B. Similarly, the first positioner PM andanother position sensor PS1 can be used to accurately position thepatterning device (e.g. mask) MA with respect to the path of theradiation beam B. Patterning device (e.g. mask) MA and substrate W maybe aligned using patterning device alignment marks M1, M2 and substratealignment marks P1, P2.

The depicted apparatus 1000 could be used in at least one of thefollowing modes:

1. In step mode, the support structure (e.g. patterning device table) MTand the substrate table WT are kept essentially stationary, while anentire pattern imparted to the radiation beam is projected onto a targetportion C at one time (i.e. a single static exposure). The substratetable WT is then shifted in the X and/or Y direction so that a differenttarget portion C can be exposed.

2. In scan mode, the support structure (e.g. patterning device table) MTand the substrate table WT are scanned synchronously while a patternimparted to the radiation beam is projected onto a target portion C(i.e. a single dynamic exposure). The velocity and direction of thesubstrate table WT relative to the support structure (e.g. patterningdevice table) MT may be determined by the (de-)magnification and imagereversal characteristics of the projection system PS.

3. In another mode, the support structure (e.g. patterning device table)MT is kept essentially stationary holding a programmable patterningdevice, and the substrate table WT is moved or scanned while a patternimparted to the radiation beam is projected onto a target portion C. Inthis mode, generally a pulsed radiation source is employed and theprogrammable patterning device is updated as required after eachmovement of the substrate table WT or in between successive radiationpulses during a scan. This mode of operation can be readily applied tomaskless lithography that utilizes programmable patterning device, suchas a programmable mirror array of a type as referred to above.

FIG. 21 shows the apparatus 1000 in more detail, including the sourcecollector module SO, the illumination system IL, and the projectionsystem PS. The source collector module SO is constructed and arrangedsuch that a vacuum environment can be maintained in an enclosingstructure 220 of the source collector module SO. An EUV radiationemitting plasma 210 may be formed by a discharge produced plasma source.EUV radiation may be produced by a gas or vapor, for example Xe gas, Livapor or Sn vapor in which the very hot plasma 210 is created to emitradiation in the EUV range of the electromagnetic spectrum. The very hotplasma 210 is created by, for example, an electrical discharge causingat least partially ionized plasma. Partial pressures of, for example, 10Pa of Xe, Li, Sn vapor or any other suitable gas or vapor may berequired for efficient generation of the radiation. In an embodiment, aplasma of excited tin (Sn) is provided to produce EUV radiation.

The radiation emitted by the hot plasma 210 is passed from a sourcechamber 211 into a collector chamber 212 via an optional gas barrier orcontaminant trap 230 (in some cases also referred to as contaminantbarrier or foil trap) which is positioned in or behind an opening insource chamber 211. The contaminant trap 230 may include a channelstructure. Contamination trap 230 may also include a gas barrier or acombination of a gas barrier and a channel structure. The contaminanttrap or contaminant barrier 230 further indicated herein at leastincludes a channel structure, as known in the art.

The collector chamber 211 may include a radiation collector CO which maybe a so-called grazing incidence collector. Radiation collector CO hasan upstream radiation collector side 251 and a downstream radiationcollector side 252. Radiation that traverses collector CO can bereflected off a grating spectral filter 240 to be focused in a virtualsource point IF along the optical axis indicated by the dot-dashed line‘O’. The virtual source point IF is commonly referred to as theintermediate focus, and the source collector module is arranged suchthat the intermediate focus IF is located at or near an opening 221 inthe enclosing structure 220. The virtual source point IF is an image ofthe radiation emitting plasma 210.

Subsequently the radiation traverses the illumination system IL, whichmay include a facetted field mirror device 22 and a facetted pupilmirror device 24 arranged to provide a desired angular distribution ofthe radiation beam 21, at the patterning device MA, as well as a desireduniformity of radiation intensity at the patterning device MA. Uponreflection of the beam of radiation 21 at the patterning device MA, heldby the support structure MT, a patterned beam 26 is formed and thepatterned beam 26 is imaged by the projection system PS via reflectiveelements 28, 30 onto a substrate W held by the substrate table WT.

More elements than shown may generally be present in illumination opticsunit IL and projection system PS. The grating spectral filter 240 mayoptionally be present, depending upon the type of lithographicapparatus. Further, there may be more mirrors present than those shownin the figures, for example there may be 1-6 additional reflectiveelements present in the projection system PS than shown in FIG. 20.

Collector optic CO, as illustrated in FIG. 20, is depicted as a nestedcollector with grazing incidence reflectors 253, 254 and 255, just as anexample of a collector (or collector mirror). The grazing incidencereflectors 253, 254 and 255 are disposed axially symmetric around theoptical axis O and a collector optic CO of this type may be used incombination with a discharge produced plasma source, often called a DPPsource.

Alternatively, the source collector module SO may be part of an LPPradiation system as shown in FIG. 22. A laser LA is arranged to depositlaser energy into a fuel, such as xenon (Xe), tin (Sn) or lithium (Li),creating the highly ionized plasma 210 with electron temperatures ofseveral 10's of eV. The energetic radiation generated duringde-excitation and recombination of these ions is emitted from theplasma, collected by a near normal incidence collector optic CO andfocused onto the opening 221 in the enclosing structure 220.

The embodiments may further be described using the following clauses:

-   1. A method for determining a stack configuration for a substrate    subjected a patterning process, the method comprising:

obtaining (i) measurement data of a stack configuration with locationinformation on a printed substrate, (ii) a substrate model configured topredict a stack characteristic based on a location of the substrate, and(iii) a stack map including a plurality of stack configurations based onthe substrate model;

determining, by a hardware computer system, values of model parametersof the substrate model based on a fitting between the measurement dataand the plurality of stack configurations of the stack map; and

predicting, by the hardware computer system, an optimum stackconfiguration at a particular location based on the substrate modelusing the values of the model parameters.

-   2. The method of clause 1, wherein the substrate model includes one    or more models corresponding to the stack characteristic of one or    more layers of the substrate.-   3. The method of any of clauses 1-2, wherein the substrate model is    expressed in Cartesian coordinates having a first set of model    parameters, and/or in polar coordinates having a second set of model    parameters.-   4. The method of clause 3, wherein the second set of model    parameters is associated with Zernike polynomials.-   5. The method of any of clauses 1-4, wherein the stack configuration    comprises a plurality of layers of the substrate, wherein each layer    is associated with the stack characteristics.-   6. The method of any of clauses 1-5, wherein the stack    characteristic is a thickness of a layer of the substrate, a    critical dimension of a feature of the substrate, and/or a distance    between adjacent features of the substrate.-   7. The method of any of clauses 1-5, wherein the stack    characteristic is a difference in a thickness of a layer and a    selected thickness of the layer.-   8. The method of clause 1, wherein the determining the values of the    model parameters of the substrate model is an iterative process, an    iteration comprising:

generating the stack map having the plurality of stack configurationsbased on simulation of the substrate model and a patterning process;

predicting intermediate values of model parameters based on anoptimization algorithm; and

fitting the measurement data and the plurality of stack configurationsof the stack map such that a cost function is reduced.

-   9. The method of clause 1, wherein the patterning process comprises    a design for control process configured to automatically predict the    stack configuration using the substrate model as perturbations.-   10. The method of any of clauses 1 or 8, wherein the measurement    data comprises a metrology recipe used for measurement of one or    more stack characteristics of the stack configuration at the    particular location on the substrate.-   11. The method of clause 10, further comprising converting    measurement data from a Cartesian coordinates to polar coordinates    using Zernike based conversion model.-   12. A method for determining optimum values of model parameters of a    model configured to predict a characteristic of a patterning    process, the method comprising:

obtaining (i) initial values including a starting point and a searchregion of the model parameters, (ii) measurement data corresponding tothe characteristic of the patterning process, (iii) a predictedcharacteristic using the initial values of the model parameter and themeasurement data, and (iv) an objective function, wherein the objectivefunction comprises a first term related to a fit level, and a secondterm representing a penalty; and

determining, by a hardware computer system, the values of the modelparameter based on the starting point, the search region, the fit levelbetween the model and the measurement data such that the objectivefunction is reduced.

-   13. The method of clause 12, wherein the characteristic of the    patterning process is a stack characteristic.-   14. The method of clause 13, wherein the stack characteristic is a    substrate thickness, a thickness deviation, an overlay, and/or an    alignment.-   15. The method of any of clauses 13-14, wherein the model is a    substrate model representing the stack characteristic.-   16. The method of clause 15, wherein the substrate model has a    parabolic form.-   17. The method of any of clauses 12-16, wherein the search region is    defined by a radius with the starting point as a center, wherein the    radius is a distance from a center.-   18. The method of any of clauses 12-17, wherein the fit level is a    difference between a predicted characteristic and the measurement    data.-   19. The method of any of clauses 12-18, wherein determining the    values of the model parameter is an iterative process, wherein an    iteration comprises:

determining a number of sample points to be selected from the searchregion based on a number of model parameters and a size of the searchregion;

fitting the model and the measurement data based on the selected samplepoints;

determining a fit level based on the fitting;

evaluating the objective function comprising the fit level;

evaluating a fit quality based on the objective function; and

updating the starting point and the search region based on the fitquality such that the objective function is reduced.

-   20. The method of clause 19, wherein the updating the starting point    and the search region comprises selecting a new starting point and    increasing the search region, in response to the fit quality    breaching a first threshold.-   21. The method of clause 19, wherein the updating the starting point    and the search region comprises selecting a new starting point, in    response to the fit quality breaching a second threshold.-   22. The method of clause 19, wherein updating the starting point and    the search region comprises decreasing a size of the search region,    in response to the fit quality breaching a third threshold.-   23. The method of clause 19, wherein the fitting is based on the    objective function comprising a cost function of second order.-   24. The method of any of clauses 12-23, wherein the objective    function comprises:

a first penalty term configured to maintain a positive value ofcoefficients of second order terms of the cost function; and/or

a second penalty term associated with a distance between predictedcharacteristic and the measurement data.

-   25. A computer program product comprising a non-transitory computer    readable medium having instructions recorded thereon, the    instructions when executed by a computer implementing the method of    any of the above clauses.

The concepts disclosed herein may simulate or mathematically model anygeneric imaging system for imaging sub wavelength features, and may beespecially useful with emerging imaging technologies capable ofproducing increasingly shorter wavelengths. Emerging technologiesalready in use include EUV (extreme ultra violet), DUV lithography thatis capable of producing a 193 nm wavelength with the use of an ArFlaser, and even a 157 nm wavelength with the use of a Fluorine laser.Moreover, EUV lithography is capable of producing wavelengths within arange of 20-5 nm by using a synchrotron or by hitting a material (eithersolid or a plasma) with high energy electrons in order to producephotons within this range.

While the concepts disclosed herein may be used for imaging on asubstrate such as a silicon wafer, it shall be understood that thedisclosed concepts may be used with any type of lithographic imagingsystems, e.g., those used for imaging on substrates other than siliconwafers.

The descriptions above are intended to be illustrative, not limiting.Thus, it will be apparent to one skilled in the art that modificationsmay be made as described without departing from the scope of the claimsset out below.

1. A non-transitory computer-readable medium comprising instructionsstored therein that, when executed by one or more processors, cause theone or more processors to at least: obtain (i) initial values includinga starting point and a search region of model parameters of a modelconfigured to predict a characteristic of a patterning process, (ii)measurement data corresponding to the characteristic of the patterningprocess, (iii) a predicted characteristic using the initial values ofthe model parameter and the measurement data, and (iv) an objectivefunction, wherein the objective function comprises a first term relatedto a fit level, and a second term representing a penalty; and determinevalues of the model parameters based on the starting point, the searchregion, and the objective function.
 2. The medium of claim 1, whereinthe characteristic of the patterning process is a stack characteristic,wherein the model is a substrate model representing the stackcharacteristic, and wherein the fit level is indicative of a differencebetween a predicted characteristic and the measurement data.
 3. Themedium of claim 2, wherein the stack characteristic is a substratethickness, a thickness deviation, an overlay, and/or an alignment. 4.The medium of claim 1, wherein the instructions are further configuredto cause the one or more processors to predict an optimum stackconfiguration at a particular location based on the substrate modelusing the values of the model parameters.
 5. The medium of claim 2,wherein the substrate model includes one or more models corresponding tothe stack characteristic of one or more layers of the substrate.
 6. Themedium of claim 4, wherein the substrate model has a parabolic form. 7.The medium of claim 1, wherein the search region is defined by a radiuswith the starting point as a center, wherein the radius is a distancefrom a center.
 8. The medium of claim 1, wherein the instructionsconfigured to cause the one or more processors to determine the valuesof the model parameters are further configured to cause the one or moreprocessors to: determine a number of sample points to be selected fromthe search region based on a number of model parameters and a size ofthe search region; and update the starting point and/or the searchregion based on the objective function.
 9. The medium of claim 1,wherein the determination of the values of the model parameterscomprises an iterative process, wherein an iteration comprises:determination of a number of sample points to be selected from thesearch region based on a number of model parameters and a size of thesearch region; fitting of the model and the measurement data based onthe selected sample points; determination of a fit level based on thefitting; evaluation of the objective function comprising the fit level;evaluation of a fit quality based on the objective function; andupdating of the starting point and the search region based on the fitquality such that the objective function is reduced.
 10. The medium ofclaim 9, wherein the updating of the starting point and the searchregion comprises selection of a new starting point and adjustment of thesearch region, in response to the fit quality breaching one or morethresholds.
 11. The medium of claim 1, wherein the objective functioncomprises: a first penalty term configured to maintain a positive valueof coefficients of second order terms of the cost function; and/or asecond penalty term associated with a distance between a predictedcharacteristic and the measurement data.
 12. The medium of claim 1,wherein the determination of the values of the model parameters is aniterative process, wherein an iteration comprises: generation of a stackmap having a plurality of stack configurations based on simulation ofthe substrate model and a patterning process; prediction of intermediatevalues of model parameters based on an optimization algorithm; andfitting of the measurement data and the plurality of stackconfigurations of the stack map such that a cost function is reduced.13. The medium of claim 1, wherein the patterning process comprises adesign for control process configured to automatically predict a stackconfiguration using the substrate model as perturbations.
 14. The mediumof claim 1, wherein the measurement data comprises a metrology recipeused for measurement of one or more stack characteristics of a stackconfiguration at a particular location on the substrate.
 15. The mediumof claim 1, wherein the instructions are further configured to cause theone or more processors to convert measurement data from a Cartesiancoordinates to polar coordinates using a Zernike based conversion model.16. A non-transitory computer-readable medium comprising instructionsstored therein that, when executed by one or more processors, cause theone or more processors to at least: obtain (i) measurement data of astack configuration, with location information, for a substratesubjected to a patterning process, (ii) a substrate model configured topredict a stack characteristic based on a location associated with thesubstrate, and (iii) a stack map including a plurality of stackconfigurations based on the substrate model; determine values of modelparameters of the substrate model based on a fitting between themeasurement data and the plurality of stack configurations of the stackmap; and predict a stack configuration at a particular location based onthe substrate model using the values of the model parameters.
 17. Themedium of claim 16, wherein the substrate model includes one or moremodels corresponding to the stack characteristic of one or more layersof the substrate.
 18. The medium of claim 16, wherein the substratemodel is expressed in Cartesian coordinates having a first set of modelparameters, and/or in polar coordinates having a second set of modelparameters.
 19. The medium of claim 16, wherein the stack configurationcomprises a plurality of layers of the substrate, wherein each layer isassociated with a stack characteristic.
 20. The medium of claim 16,wherein the stack characteristic is a thickness of a layer of thesubstrate, a critical dimension of a feature of the substrate, adistance between adjacent features of the substrate, and/or a differencebetween a thickness of a layer and a selected thickness of the layer.